# Find the limit $L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}}$

Find the limit following:

$$L=\lim_{ _{\Large {n\to \infty}}}\:\sqrt{\frac{1}{2}+\sqrt[\Large 3]{\frac{1}{3}+\cdots+\sqrt[\Large n]{\frac{1}{n}}}}$$

P.S

I tried to find the value of $\:L$, but I found myself stuck into the abyss of incertitude.

Thus, any help to get me out of this rift is more than welcome!

• Possibly related: math.stackexchange.com/questions/576110/… – abiessu Nov 26 '13 at 18:16
• The numerical value is 1.2722249619362552835210450521628613228181075332403 , according to PARI. Using the inverse symbolic calculator, I found no closed expression. – Peter Nov 26 '13 at 18:28
• n=10000;u=(1/n)^(1/n);while(n>2,n=n-1;u=(u+1/n)^(1/n));print(u) – Peter Nov 26 '13 at 18:51
• You could see if Landau's Algorithm (for denesting radicals) is of any help, but my guess is there isn't a "nice" expression for what you have here... – Benjamin Dickman Nov 26 '13 at 20:06
• I guess its limit is 1. Because it is increasing sequence which is bounded(maybe) by a number less than 2. But I don't know how to prove it! – Hamid Shafie Asl Jun 11 '14 at 11:20

I like how Yiorgos S. Smyrlis approached to find upper limit of $L$. In similar way, you can easily observe $\sqrt{\frac{1}{2}+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{2}+\cdots}}} > L$ and lets assume that it converges to some constant $c$. Now, we can write ,

$\sqrt{\frac{1}{2}+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{2}+\cdots}}} = c$

Squaring on both sides,

$\frac{1}{2}+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{2}+\cdots}}} = c^2$

Which is nothing but,

$\frac{1}{2}+c = c^2$

Solving above quadratic expression, value of $c$ will be $\dfrac{1+\sqrt{3}}{2}$ . Thus we get slightly improved upper bound for $L$ as $L < \dfrac{1+\sqrt{3}}{2}$

• That's even better! I wonder if it is possible to get even tighter bound... – Aron D'souza Feb 20 '16 at 11:51
• The next bound is approximately $L<1.272871$ – Yuriy S Feb 20 '16 at 11:54
• Add it as an answer then. – Aron D'souza Feb 20 '16 at 11:55

Using Aron D'souza's idea further we can get:

$$L^2-\frac{1}{2}< \sqrt[3]{\frac{1}{3}+\sqrt[3]{\frac{1}{3}+\dots}}$$

$$\sqrt[3]{\frac{1}{3}+\sqrt[3]{\frac{1}{3}+\dots}}=\frac{1}{2} \left(1+\sqrt{\frac{7}{3}} \right)$$

$$L<\sqrt{1+\frac{1}{2} \sqrt{\frac{7}{3}}}=1.328067$$

To find the next bound we will need to solve:

$$c^4-c-\frac{1}{4}=0$$

The exact solution is too complex to write here (see Wolframalpha), so I'll just write it numerically:

$$\sqrt[4]{\frac{1}{4}+\sqrt[4]{\frac{1}{4}+\dots}}=1.0723501510383$$

The bound will become:

$$L<\sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+1.0723501510383}}=1.272871$$

Which is three correct digits of the numerical value of the limit.

To make my answer more complete, the exact value of $c_4$ is:

$$c=\frac{1}{2} \left( b+\sqrt{\frac{2}{b}-b^2} \right)$$

$$b=\sqrt{ \sqrt[3]{ \frac{a}{18} }-\sqrt[3]{ \frac{2}{3a} } }$$

$$a=9+\sqrt{93}$$

And solving the quintic equation:

$$c^5-c-\frac{1}{5}=0$$

We get the upper bound for the limit with four correct digits:

$$L<1.272282$$

Taking into account the corresponding lower boundary:

$$L>\sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\sqrt[4]{\frac{1}{4}+\sqrt[5]{\frac{1}{5}+\sqrt[6]{\frac{1}{6}}}}}}=1.271035$$

We see that truncating the limit gives less accurate solutions than the method in this answer.

However, truncating at $\frac{1}{7}$ we can finally get very good boundaries:

$$1.27207<L<1.27228$$

This is a partial result:

The underlying sequence is increasing and upper bounded by $\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}=\dfrac{1+\sqrt{5}}{2}=\phi$. Thus the limit exists and it is less than $\phi$.

Actually, there is another way which gives better upper boundary. I'm posting it as a separate answer because of the size.

First we notice that:

$$\lim_{n \to \infty} \left( \frac{1}{n} \right)^{\frac{1}{n}}=\lim_{n \to \infty} \left(1- \left(1- \frac{1}{n} \right) \right)^{\frac{1}{n}}=1$$

This is not a proof, but the fact is well known. Now let's consider the following:

$$1< \left(\frac{1}{n-1}+1 \right)^{\frac{1}{n-1}}<1+\frac{1}{(n-1)^2}$$

$$1< \left(\frac{1}{n-2}+1+\frac{1}{(n-1)^2} \right)^{\frac{1}{n-2}}<1+\frac{1}{(n-2)^2}+\frac{1}{(n-2)(n-1)^2}$$

On the next step we get:

$$\dots 1+\frac{1}{(n-3)^2}+\frac{1}{(n-3)(n-2)^2}+\frac{1}{(n-3)(n-2)(n-1)^2}$$

In the end we obtain the following inequality:

$$\lim_{ _{\Large {n\to \infty}}}\:\sqrt{\frac{1}{2}+\sqrt[\Large 3]{\frac{1}{3}+\cdots}}<1+\sum^{\infty}_{k=2} \frac{1}{k ~k!}=Ei(1)-\gamma=1.3179$$

We can increase precision by moving the truncated series under the radical (and we should not forget to get rid of $2$ in every denominator):

$$1+2\sum^{\infty}_{k=3} \frac{1}{k ~k!}=1+2(Ei(1)-\gamma-1-1/4)=1.135804$$

$$L < \sqrt{\frac{1}{2}+1.135804}=1.27899$$

$$1+2\cdot 3 \sum^{\infty}_{k=4} \frac{1}{k ~k!}=1+6(Ei(1)-\gamma-1-1/4-1/18)=1.074080$$

$$L < \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+1.074080}}=1.27305$$

Now for the lower boundary the better estimation would be:

$$L > \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+1}}=1.26517$$

This is not ideal, but much more accurate than just truncating the sequence.


# Do both versions (invariant and primary) of the Fundamental Theorem for Finitely Generated Abelian Groups hold at the same time?

So there are the two versions of the Fundamental Theorem for Finitely Generated Abelian Groups (FTFGAG). I take the following from A First Course in Abstract Algebra by Fraleigh. The first is as follows:

FTFGAG 1: Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups in the form $$G \cong \mathbb{Z}_{p_1^{r_1}} \times \cdots \times \mathbb{Z}_{p_n^{r_n}} \times \mathbb{Z} \times \cdots \times \mathbb{Z}$$ where $p_i$ are primes (not necessarily distinct) and $r_i$ are positive integers.

But then we also have the second version:

FTFGAG 2: Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups in the form $$G \cong \mathbb{Z}_{m_1} \times \cdots \times \mathbb{Z}_{m_r} \times \mathbb{Z} \times \cdots \times \mathbb{Z}$$ where $m_1 | m_2 | \cdots | m_r$.

My question is whether these two hold at all times? So say for $\mathbb{Z}_{20}$, do we have $\mathbb{Z}_{20} \cong \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_5 \cong \mathbb{Z}_4 \times \mathbb{Z}_5 \cong \mathbb{Z}_2 \times \mathbb{Z}_{10}$, even though $10$ is not a power of a prime (though of course $2|10$)?

I've also seen statements of the Chinese Remainder Theorem (CRT) which say that $\mathbb{Z}_{nm} \cong \mathbb{Z}_n \times \mathbb{Z}_m$ if and only if $\gcd(n,m)=1$. Does this not contradict $\mathbb{Z}_{20} \cong \mathbb{Z}_2 \times \mathbb{Z}_{10}$ from above? Or is treating $\mathbb{Z}_n$ and $\mathbb{Z}_m$ as rings in the CRT what makes the situation different? I guess what I'm asking in a nutshell is: why don't the primary and invariant forms of the structure theorems for abelian groups (and also modules over PIDs, etc) contradict each other?

Edit: So it seems that I didn't see this question which basically answers mine. CRT gives us a way of decomposition but FTFGAG only tells us that some decomposition is always possible. So for FTFGAG 1, $\mathbb{Z}_4 \times \mathbb{Z}_5$ suffices, for FTFGAG 2, $\mathbb{Z}_{20}$ suffices, and for the CRT $\mathbb{Z}_{20} \cong \mathbb{Z}_4 \times \mathbb{Z}_5$ works.

• Yes they are both correct theorems so they both hold for all finitely generated abelian groups. May 27, 2018 at 13:02
• Thanks, would you be able to shed some light on why the stuff in the CRT paragraph is/isn't a contradiction? May 27, 2018 at 13:04
• It is not true that $\mathbb{Z}_{20}\simeq\mathbb{Z}_2\times\mathbb{Z}_{10}$. May 27, 2018 at 13:09
• @MichaelBurr Indeed, I have just seen the (~duplicate) question which I linked in my edit. May 27, 2018 at 13:12

For your example, $\mathbb{Z}_{20}$,

• The first decomposition gives you $\mathbb{Z}_{20}\simeq\mathbb{Z}_{2^2}\times\mathbb{Z}_5$.

• The second decomposition gives you $\mathbb{Z}_{20}\simeq\mathbb{Z}_{20}$. In other words, this one doesn't break up the abelian group at all.

• You can note that $\mathbb{Z}_{20}\not\simeq\mathbb{Z}_2\times\mathbb{Z}_{10}$ since $\mathbb{Z}_{20}$ has an element of order $20$ while the maximum order of an element of $\mathbb{Z}_2\times\mathbb{Z}_{10}$ is $10$.

The idea of the statements is that it is possible to write it in this form, not that all products of the given form are isomorphic to the given group.

A more interesting example might be given by $$\mathbb{Z}_4\times\mathbb{Z}_6\times\mathbb{Z}_5.$$

• The first decomposition gives you $$\mathbb{Z}_4\times\mathbb{Z}_6\times\mathbb{Z}_5\simeq \left(\mathbb{Z}_{2^2}\right)\times\left(\mathbb{Z}_2\times\mathbb{Z}_3\right)\times\mathbb{Z}_5\simeq \mathbb{Z}_2\times\mathbb{Z}_{2^2}\times\mathbb{Z}_3\times\mathbb{Z}_5.$$

• On the other hand, the second decomposition gives you $$\mathbb{Z}_4\times\mathbb{Z}_6\times\mathbb{Z}_5\simeq\mathbb{Z}_2\times\mathbb{Z}_{2^2\cdot 3\cdot 5}=\mathbb{Z}_2\times\mathbb{Z}_{60}.$$ The group on the right combines the highest powers of each prime that appear in the fully expanded product of the first decomposition, then you work your way down by induction.

• "The group on the right combines the highest powers of each prime that appear in the fully expanded product of the first decomposition, then you work your way down by induction." I really like that way of thinking about it, thanks! May 27, 2018 at 13:23

You are correct about the Chinese remainder theorem indeed $\mathbb{Z}_{20} \not\cong \mathbb{Z}_2 \times \mathbb{Z}_{10}$. The problem with applying the 2nd version in this manner is the rank uniquely determines the group. So for example $\mathbb{Z}_{20}$ is rank 1, that is it already is written in that form (trivially)


# Calculus / find the value of $x$ so that $f ''(x)=0$

Let $f(x)= 10xe^x$

$(a)$ Find the exact value of $x$ so that $f ''(x) = 0$.

I tried:

\begin{align}f'(x)& = 10e^x\\f''(x)&= e^x\end{align} but at that point, the $f''(x)$ would never be a zero. So what is my mistake?

$(b)$ For what interval is $f(x)$ concave up?

I wonder how to know the concavity after knowing the equation

• One mistake is not using the product rule to evaluate $(10xe^x)'$. – David Mitra Jun 18 '14 at 23:45
• Once you have the correct second derivative function, $\ f(x) \$ is "concave upward" wherever $\ f \ ''(x) \ > \ 0 \$ and "concave downward" wherever $\ f \ ''(x) \ < \ 0 \$ . Keep in mind that the exponential factor $\ e^x \$ is always positive. – colormegone Jun 18 '14 at 23:49
• @DavidMitra thanks for the reminder – John Jun 18 '14 at 23:50

(a) $f'(x)=10e^x + 10xe^x$.

$f''(x)=10e^x + 10e^x + 10xe^x = 20e^x + 10xe^x =10e^x(2+x)$

So, at $x=-2$ you have $f''(x)=0$.

(b) Hint: If $f''(x)>0$, $f$ is concave up at $x$, and if $f''(x)<0$, $f$ is concave down at $x$.

• Thaanks !What did you do to go from f'(x) to f''(x)? – John Jun 18 '14 at 23:49
• I computed it using the product rule. – Twink Jun 18 '14 at 23:50
• @John, derivation using the product rule. $f''(x) = (10e^x + 10xe^x)' = 10 (e^x)' + 10(x)'e^x + 10x(e^x)' = 10e^x + 10 e^x + 10xe^x = 10(2+x)e^x$ – Graham Kemp Jun 18 '14 at 23:52

$(a)$ Since $f(x)=10x\cdot e^x$ we will use the product rule to obtain $f'(x)$ and $f''(x)$. So $f'(x)=10x\cdot e^x+10e^x$ and $f''(x)=10x\cdot e^x+20e^x$. Set $f''(x)=0$. So $10x\cdot e^x+20e^x=0$ implies that $10e^x(x+2)=0$. We know that $10e^x$ is never $0$ and so $x=-2$ will make $f''(x)=0$.

$(b)$ If $x<-2$, then the function is concave down. If $x>-2$, then the function is concave up.


# Arbitrary union definition in set theory

I am reading Enderton's "Elements of Set Theory". He defines the union operation as $$\cup A = \{ x \;|\; x \text{ belongs to some member of } A\} = \{x \;|\; (\exists b \in A) x \in b\}$$ Maybe I am missing a subtle point from earlier on in the text, but what if the set in question is not a "set of sets"? For example, if $$A = \{1,2,3\}$$, then is $$\cup A = \emptyset$$?

• In systems of set theory such as ZF, every set is "a set of sets". – Lord Shark the Unknown Mar 30 at 19:11
• So then is my $A$ really $\{\{1\},\{2\},\{3\}\}$? – theQman Mar 30 at 19:12
• No, it's really $\{1,2,3\}$. – Lord Shark the Unknown Mar 30 at 19:13
• See Enderton's chapter on "Natural Numbers" to see the usual way of representing positive integers as sets. – Lord Shark the Unknown Mar 30 at 19:20
• Even without the "everything-is-a-set" context, we can still make sense of this by carefully writing down the definition: see this earlier question. – Noah Schweber Mar 30 at 19:32

When you're working in a system built out of the set theory, everything is a set. I assume the book will cover this later on, but one way of "building natural numbers" out of sets is called Von Neumann ordinals and the construction is as follows:

Let $$0 \equiv \emptyset$$. Let the number $$n \equiv n - 1 \cup \{ n - 1\}$$. That is, under this system: \begin{align*} &1 \equiv 0 \cup \{ 0 \} = \emptyset \cup \{\emptyset\} = \{\emptyset\} \\ &2 \equiv 1 \cup \{1 \} = \{\emptyset\} \cup \{\{\emptyset\}\} = \{\emptyset, \{\emptyset\}\} \\ &3 \equiv 2 \cup \{2\} = \dots = \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\} \end{align*}

The point of the representation is that 0 is a set with zero elements, 1 is a set with 1 element, 2 is a set with 2 elements and so on. So we can define the "size of a set" as "the natural number it is in bijection with".

This goes down a rabbit hole, and we can then ask "well, now that I have the natural numbers, how do I define addition? Multiplication? What about the integers? rationals? reals?"

We can construct all of these things, and a set theory book usually describes these encodings in one of the later chapters.

But the point is that at this level, all the "stuff of math" is sets, and you can always write expressions such as $$1 \ \cup 2 \cup 3 = \{\emptyset\} \cup \{ \emptyset, \{\emptyset\}\} \cup \{ \emptyset, \{\emptyset\},\{ \emptyset, \{\emptyset\}\} \} = \{ \emptyset, \{\emptyset\},\{ \emptyset, \{\emptyset\}\}\} = 3$$!

• +1.... In the most-favored system of set theory, not only is everything (that can be said to exist) a set, but in the formal language there is no word or definition of "set". Things ( with various properties that $can$ be stated in the language ) are merely asserted to exist or not exist. – DanielWainfleet Mar 30 at 19:39
• Indeed, thanks! much appreciated – Siddharth Bhat Mar 30 at 20:18

As Lord Shark the Unknown indicates, in nearly all systems of set theory, every object under the sun is a set. In particular, the elements of any set are, in turn, sets themselves. For the specific example you give, i.e. $$\cup \{ 1,2,3 \},$$ we need to be able to describe the natural numbers as sets. On page 67 of Enderton's text, the construction is outlined. He defines \begin{align} 0 &= \varnothing \\ 1 &= \{0\} = \{ \varnothing \} \\ 2 &= \{0,1\} = \{ \varnothing, \{\varnothing\} \} \\ 3 &= \{0,1,2\} = \{ \varnothing, \{\varnothing\}, \{ \varnothing, \{\varnothing\} \}\}, \end{align} and so on. The "and so on" is explained in somewhat more detail in chapter 4 of the text (starting on page 66). Note that we could write $$2 = \{\varnothing, \{\varnothing\} \}$$ over and over again, but it is likely easier to see what is going on if we work one level of abstraction higher. That is, $$2 = \{0,1\}$$. Once we accept this abstraction, we have $$\cup\{1,2,3\} = 1 \cup 2 \cup 3 = \{0\} \cup \{0,1\} \cup \{0,1,2\} = \{0,1,2\} = 3.$$

• I see. Maybe I am getting ahead of myself since I am only in chapter 2. But I guess the construction you provided is pretty natural since at this point the only set that we can use as a building block is $\emptyset$ (Empty Set Axiom)? Then we can use "pairing" to form the singleton $\{\emptyset\}$, which is defined to be $\{\emptyset, \emptyset\}$? – theQman Mar 31 at 2:09


Question

# If two concentric circles are drawn with centre in first quadrant, one touches the X-axis and Y-axis and the other passes through the origin, the radius of the smaller circle is $$'r'$$, then the centre is

A
(2r,r)
B
(r,2r)
C
(r,r)
D
none of these

Solution

## The correct option is C $$(r,r)$$Let $$(a,b)$$ be the center of the circles Given that one of the circles touches both the axes and the other passes through origin. Smaller circle touches the axes and Bigger circle passes through $$O$$. Given radius of smaller circle is $$r$$ and $$X,Y$$ axes are tangents Which implies radius = perpendicular distance from center to tangent $$(y =0 , x = 0)$$ $$r = \dfrac{|a|}{\sqrt{1^2 + 0}} , r = \dfrac{|b|}{\sqrt{1^2 + 0}}$$ $$\implies a = r, b = r$$ $$\implies (a, b) = (r,r) = center$$Maths

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## The Plane in R^3

Prerequisites: Euclidean Space and Vectors

### Suppose we have a plane in ${\Bbb R}^{3}$ which contains the point $(0,0,0)$ but does not intersect the axes at any other point. How many octants does the plane intersect?

Let's start by analyzing the 2-dimensional analog, then come back to the 3-dimensional problem with an algebraic approach and a geometric interpretation. Finally, we'll try to generalize the answer to higher dimensions.

#### Lines in the Plane ${\Bbb R}^{2}$

The 2-dimensional analog of a plane in ${\Bbb R}^{3}$ containing the origin is a line in ${\Bbb R}^{2}$ containing the origin. For those familiar with linear algebra, the former is a vector subspace of ${\Bbb R}^{3}$ of dimension 2 (i.e. co-dimension 1), and the latter is a vector subspace of ${\Bbb R}^{2}$ of dimension 1 (still co-dimension 1). If you aren't familiar with linear algebra, ignore that sentence and just keep reading.

The equation of a line in ${\Bbb R}^{2}$ is $y = mx + b$, where $m$ is the slope ("rise over run," or change in $y$ per change in $x$), and $b$ is the $y$-intercept (where the line hits the $y$-axis). A point $(x_{0},y_{0})$ is on the line if it satisfies the equation, i.e. if the equality $y_0 = mx_0 + b$ is true.

A line passing through the origin has a $y$-intercept of 0, thus $b=0$ and the equation is simply $y=mx$. Now, if $m$ is 0, then the line would just go along the $x$-axis, so if it doesn't touch the axis, we must have $m \neq 0$, so either $m > 0$ or $m < 0$. In the case where $m>0$, a positive $x$ value gives a positive $y$, and a negative $x$ gives a negative $y$, so the line would pass through quadrants 1 and 3 (top-right and bottom-left). In the case where $m<0$, the same analysis shows that the line passes through quadrants 2 and 4 (top-left and bottom-right). So the line passes through 2 quadrants.

The $y = mx + b$ formulation works fine for this question, but if we want to put $x$ and $y$ on more even footing (this will come in handy in the higher-dimensional cases so that we don't always have to solve for one of the variables), we can use the other form of the equation of a line, $ax + by = c$, where $a,b,c \in {\Bbb R}$ are constants. In order to have the origin on the line, we must have $c=0$, because the point $(0,0)$ is on the line, and thus $a(0)+b(0) = 0 = c$. so the equation is simply $ax+by=0$. If either $a$ or $b$ were zero, the line would just be one of the axes, so we must have $a,b \neq 0$. Whether they are positive or negative, you can work out by plugging in positive or negative $x$ and $y$ values that the line will either pass through quadrants 1 and 3 or 2 and 4.

For example, if $a,b>0$, then if $x>0$, we must have $y<0$ in order to have $ax+by=0$. Similarly, if $x<0$, then $y$ would have to be positive. So the line passes through quadrants 2 and 4. We get the same result in the case where $a,b<0$. If $a$ and $b$ have opposite signs, then the line will pass through quadrants 1 and 3.

Notice that the line does not pass through the quadrant containing the point $(a,b)$. The vector $(a,b)$ is actually perpendicular to the line $ax+by=0$, and we can see that from the fact that the equation can be rewritten as ${\bf n} \cdot {\bf x} = 0$ where ${\bf n} = (a,b)$ and ${\bf x} = (x,y)$ (recall that two vectors are perpendicular if and only if their dot product is zero).

#### The Plane in Space ${\Bbb R}^{3}$

Let's go back to the ${\Bbb R}^{3}$ case, building off of the above discussion. The general equation of a plane is $ax+by+cz=d$, where $a,b,c,d \in {\Bbb R}$ are constants. In order for the plane to contain $(0,0,0)$, we must have $d=0$, so the equation is now just $ax+by+cz=0$. If any of the constants is 0, then the plane will actually look like the equation of a line from above. For example, if $c=0$, then we'd have $ax+by=0$, which would be a line in the $xy$-plane extended vertically up and down in the positive and negative $z$ directions, and in fact containing the entire $z$-axis. Can you see why (hint: show that the points on the $z$-axis, i.e. points of the form $(0,0,z)$, all satisfy the equation of the plane)?

We can also interpret the equation geometrically as follows: the equation $ax+by+cz=0$ is equivalent to ${\bf n} \cdot {\bf x} = 0$, where ${\bf n} = (a,b,c)$ and ${\bf x} = (x,y,z)$. Note that here, ${\bf n}$ and ${\bf x}$ are vectors, and their dot product is a scalar, so the $0$ on the right is a scalar zero, not the zero vector ${\bf 0} = (0,0,0)$.

As mentioned above, the dot product of two vectors is zero if and only if the vectors are perpendicular. Therefore, this equation is saying that any vector ${\bf x}$ that is perpendicular to ${\bf n}$ is on the plane. For this reason, ${\bf n}$ is called the plane's normal vector (normal is a synonym for perpendicular, as is orthogonal, which is also used frequently). In the example above where $c=0$, the plane's normal vector is $(a,b,0)$, which lies in the $xy$-plane. Thus, the $z$-axis, being orthogonal to the $xy$-plane, is contained in our plane.

Now, in order to answer the geometric question of which vectors are orthogonal to ${\bf n}$, we can look at the algebraic equation $ax+by+cz=0$.

Since the plane does not touch the axes except at the origin, we must have $a,b,c \neq 0$. As an example, let's look at the case where $a,b,c>0$. Then we can have the following combinations for $(x,y,z)$ in order to have $ax+by+cz=0$:
$$(+,+,-) \\ (+,-,+) \\ (+,-,-) \\ (-,-,+) \\ (-,+,-) \\ (-,+,+)$$ The remaining two combinations, $(+,+,+)$ and $(-,-,-)$, do not work, because then the left side of the equation would have to be positive or negative (respectively) and thus not zero.

If we grind through the algebra of the other 7 combinations for $(a,b,c)$, we see that we get 6 possibilities each time, so the plane intersects 6 of the 8 octants, and we have the answer to the problem. I'm not going to go through all the cases, because that would be quite boring, but you can see that there is a certain symmetry in the plane's equation between $(a,b,c)$ and $(x,y,z)$. Once you've solved it for the case $a,b,c>0$, you've pretty much solved it for all the cases. Can you see why? So we've got the answer- it's 6.

#### The Hyperplane in ${\Bbb R}^{n}$

${\Bbb R}^{n}$ is the $n$-dimensional analog of ${\Bbb R}^{3}$ and is the set of ordered $n$-tuples of real numbers: ${\Bbb R}^{n} = \{ (x_1,x_2,x_3,...,x_n) \ \colon \ {\scr each} \ x_i \in {\Bbb R} \}$. We can't picture this $n$-dimensional space, but we can use the same types of algebraic equations that work in ${\Bbb R}^{3}$ to analyze ${\Bbb R}^{n}$.

${\Bbb R}^{n}$ is divided into $2^n$ orthants, also known as hyperoctants or $n$-hyperoctants, based on the signs, positive or negative, of the $n$ components of a point. A 2-hyperoctant is a quadrant in ${\Bbb R}^{2}$ and a 3-hyperoctant is an octant in ${\Bbb R}^{3}$. The $x_i$-axis is the set of points where all coordinates except possibly the $i^{\scr th}$ are zero.

A hyperplane in ${\Bbb R}^{n}$ is a set $P$ of points (equivalently, vectors) that are orthogonal to a normal vector ${\bf n} = (a_1, a_2, ... a_n)$. In symbols, $P = \{ {\bf x} \in {\Bbb R}^{n} \ \colon \ {\bf n} \cdot {\bf x} = 0 \}$. For those familiar with linear algebra, the hyperplane containing the origin is a vector subspace of ${\Bbb R}^{n}$ of dimension $n-1$, i.e. co-dimension 1. A hyperplane in ${\Bbb R}^{2}$ is a line, and a hyperplane in ${\Bbb R}^{3}$ is a plane.

### How many $n$-hyperoctants does a hyperplane $P \subset {\Bbb R}^{n}$ intersect, given that it contains the origin, but does not intersect the axes at any other point?

To answer this question, we can use the discussion above from the $n=2$ and $n=3$ cases and generalize the results. We can then prove the answer is correct using induction.

When we went from $n=2$ to $n=3$, we took the equation $ax+by=0$ (i.e. the line in ${\Bbb R}^{2}$ with normal vector $(a,b)$), extended it into 3-dimensional space to make the plane whose normal vector is $(a,b,0)$, and then added a non-zero third coordinate to the normal vector to "tilt" the plane off of the $z$-axis.

Now, a hyperplane (including the line and plane in the $n=2$ and $n=3$ cases) is orthogonal to its normal vector ${\bf n}$ as well as the negative of the normal vector, $-{\bf n}$. In fact, the hyperplane is orthogonal to any scalar multiple of ${\bf n}$, but my point in mentioning $-{\bf n}$ is that the hyperplane won't intersect the $n$-hyperoctants that contains ${\bf n}$ or $-{\bf n}$.

Let's look at the case where ${\bf n}$ lies in the first $n$-hyperoctant, i.e. has all positive coordinates. As mentioned above, the other cases are pretty much the same because of the symmetries of the equation ${\bf n} \cdot {\bf x} = \sum_{i=1}^{n}{a_i x_i} = 0$, so the number of $n$-hyperoctants the hyperplane intersects is the same in all cases. In the case that the $a_i$ are positive, the hyperplane doesn't intersect the first $n$-hyperoctant or the one with all negative coordinates (whatever number we want to assign to that one).

In the ${\Bbb R}^{2}$ case, the line intersects quadrants 2 and 4. When we extended ${\bf n} = (a,b)$ to ${\bf n} = (a,b,0)$ in ${\Bbb R}^{3}$, we got a plane that contained the entire $z$-axis. The intersection of this plane with the $xy$-plane is the line $ax+by=0$, which remains the case regardless of the third coordinate of ${\bf n}$. Now, this plane intersects the octants $(+,-,+)$, $(+,-,-)$, $(-,+,+)$, and $(-,+,-)$. We took the original quadrants 2 and 4 and multiplied them by 2 to get 4 octants.

When we add a non-zero third coordinate to ${\bf n}$ (let's assume it's positive), the new plane also intersects two additional octants: $(+,+,-)$ and $(-,-,+)$. The first two coordinates of these two would have not been included in the 2-d case, but the third coordinate allows us to use those combinations and still get the equation $ax+by+cz$ to equal zero. $(+,+,+)$ and $(-,-,-)$ still don't work though.

The same logic works when going from ${\Bbb R}^{n}$ to ${\Bbb R}^{n+1}$ when $n>2$, and we can prove it by induction.

Thoerem: For $n \geq 2$, a hyperplane in ${\Bbb R}^{n}$ containing the origin, but not intersecting the coordinate axes at any other point, intersects $2^{n}-2$ $n$-hyperoctants.

The proof is a bit lengthy, but basically just formalizes the idea of extending the line in ${\Bbb R}^{2}$ into a plane in ${\Bbb R}^{3}$ and then tilting it off the $z$-axis

Proof: The base case of $n=2$ was already shown above.

For the induction step, assume the theorem is true for ${\Bbb R}^{n-1}$, and consider a hyperplane $P = \{{\bf x} \in {\Bbb R}^{n} \ \colon \ {\bf n} \cdot {\bf x}=0 \}$ where ${\bf n} = (a_1,a_2,...,a_n)$.

The equation of $P$ is $\sum_{i=1}^{n}{a_i x_i} = \sum_{i=1}^{n-1}{a_i x_i} + a_n x_n = 0$. By the induction hypothesis, the solutions to the equation $\sum_{i=1}^{n-1}{a_i x_i} = 0$ intersect $2^{n-1}-2$ $(n-1)$-hyperoctants. Let's call those solutions $P_{n-1}$, which is a hyperplane in ${\Bbb R}^{n-1}$

Take a point ${\bf x}_{0, n-1} = (x_{0,1}, x_{0,2},...,x_{0,n-1}) \in {\Bbb R}^{n-1}$ which satisfies the equation of $P_{n-1}$. If $x_{0,1}>0$, then $x_{0,1}+\epsilon>0$ as well, where $\epsilon = \frac{1}{2}|x_{0,1}|$. Similarly, if $x_{0,1}<0$, then $x_{0,1}+\epsilon<0$ as well, so the point ${\bf x}_{1,n-1} = (x_{0,1}+\epsilon, x_{0,2},...,x_{0,n-1})$ is in the same $(n-1)$-hyperoctant as ${\bf x}_{0, n-1}$. By a similar argument, so is the point ${\bf x}_{2,n-1} = (x_{0,1}-\epsilon, x_{0,2},...,x_{0,n-1})$.

Define the points ${\bf x}_{1} = (x_{0,1}+\epsilon, x_{0,2},..., x_{0,n-1}, -\frac{a_1}{a_n}\epsilon), \ {\bf x}_{2} = (x_{0,1}-\epsilon, x_{0,2},..., x_{0,n-1}, \frac{a_1}{a_n}\epsilon) \in {\Bbb R}^{n}$. Then \begin{align} {\bf n} \cdot {\bf x}_{1} &= a_1 (x_{0,1}+\epsilon) + a_2 x_{0,2} + ... + a_{n-1} x_{0,n-1} + a_n (-\frac{a_1}{a_n}\epsilon) \\[2mm] &= a_1 (x_{0,1}+\epsilon -\epsilon) + a_2 x_{0,2} + ... + a_{n-1} x_{0,n-1} \\[2mm] &= a_1 x_{0,1} + a_2 x_{0,2} + ... + a_{n-1} x_{0,n-1} = 0 \end{align}
with the final equality being true because ${\bf x}_{0,n-1} \in P_{n-1}$.

This shows that ${\bf x}_{1} \in P$. Similarly, ${\bf x}_{2} \in P$. The first $n-1$ coordinates of these two points are in the same $(n-1)$-hyperoctant as ${\bf x}_{0,n-1}$, and the $n^{\scr th}$ coordinates of ${\bf x}_{1}$ and ${\bf x}_{2}$ have opposite sign. This shows that we have kept the $2^{n-1}-2$ $(n-1)$-hyperoctants of the $(n-1)$-hyperplane when we extended it into ${\Bbb R}^{n}$ and actually multiplied them by 2 (by adding both positive and negative $n^{\scr th}$ coordinates) to get $2(2^{n-1}-2)$ = $2^{n}- 4$ $n$-hyperoctants.

We just need to show that we've also added two more $n$-hyperoctants. These are the ones where the first $n-1$ coordinates all have the same sign or all have the opposite sign as the first $n-1$ coordinates of ${\bf n}$, just like when we went from $n=2$ to $n=3$ above. Examples of solutions to the equation of $P$ that are in those 2 $n$-hyperoctants are $(a_1,a_2,...,a_{n-1},-\dfrac{1}{a_n}\sum_{i=1}^{n-1}{a_i^2})$ and $(-a_1,-a_2,...,-a_{n-1},\dfrac{1}{a_n}\sum_{i=1}^{n-1}{a_i^2})$.

So now we are up to $2^{n}-4+2 = 2^{n}-2$, so $P$ intersects at least that many $n$-hyperoctants. There are only 2 more $n$-hyperoctants, and those are the ones that contain $\pm {\bf n}$, but we already know that points in those $n$-hyperoctants cannot satisfy the equation of ${\bf n} \cdot {\bf x} = 0$, so $P$ intersects exactly $2^{n}-2$ of the $2^n$ $n$-hyperoctants, and the theorem is proved.
$\square$

Here's a diagram illustrating the objects described in the proof in the case where $n=3$ and $n-1=2$. Apologies for the low quality (I made it in MS Paint), but note that the bottom of the red plane, $P$, comes out towards the viewer, in front of the blue plane, and the top half of $P$ is behind the blue plane. The points ${\bf x}_{0,n-1}$, ${\bf x}_{1,n-1}$, and ${\bf x}_{2,n-1}$ are in in the $x_{1}x_{2}$-plane, with ${\bf x}_{1,n-1}$ in front of $P$ and ${\bf x}_{2,n-1}$ behind $P$.

2. A hypersurface is a generalization of a hyperplane in the context of manifolds. I'm not sure the question about octants would make sense in that context since a manifold need not be in ${\Bbb R}^{n}$ (though it has a map to ${\Bbb R}^{n}$ in a neighborhood of each point). Maybe there's a way to reformulate the question to apply in that context...


IntMath Home » Forum home » Methods of Integration » Taking e to both sides

# Taking e to both sides [Solved!]

### My question

Your Integration Chapter: The Basic Logarithmic Form, example 4.

Can you show how you took e to both sides to arrive at your result please?

### Relevant page

2. Integration: The Basic Logarithm Form

### What I've done so far

After integrating, t = ln 20 - ln (20-v).

You applied log laws to get t = ln(20/(20-v)).

You took "e to both sides" to get e^t = 20/20-v.

We thought by taking e to both sides the result would be the exponents equated:

t = 20/20-v.

X

Your Integration Chapter: The Basic Logarithmic Form, example 4.

Can you show how you took e to both sides to arrive at your result please?
Relevant page

2. Integration: The Basic Logarithm Form

What I've done so far

After integrating, t = ln 20 - ln (20-v).

You applied log laws to get t = ln(20/(20-v)).

You took "e to both sides" to get e^t = 20/20-v.

We thought by taking e to both sides the result would be the exponents equated:

t = 20/20-v.

## Re: Taking e to both sides

@Phinah: The opposite process of taking the "log" of something is to take "e to the power of...".

Writing the full details would be:

t = ln (20 /(20-v))

e^t = e^[ln (20/(20-v))]

e^t = 20/(20-v)

e^t = e^[(20/(20-v))] (without "ln")

t = 20/(20-v)

but that's not what we started with.

A similar situation is the case where "raising to power 2" is the opposite of "square root".

y = sqrt(x+2)

Squaring both sides gives us:

y^2 = (sqrt(x+2))^2 = x+2

The "power 2" has "undone" the sqrt operation.

Similarly, the "e to the power" operation undoes the "ln" expression in the example you are asking about.

Hope it makes sense.

X

@Phinah: The opposite process of taking the "log" of something is to take "e to the power of...".

Writing the full details would be:

t = ln (20 /(20-v))

e^t = e^[ln (20/(20-v))]

e^t = 20/(20-v)

e^t = e^[(20/(20-v))] (without "ln")

t = 20/(20-v)

but that's not what we started with.

A similar situation is the case where "raising to power 2" is the opposite of "square root".

y = sqrt(x+2)

Squaring both sides gives us:

y^2 = (sqrt(x+2))^2 = x+2

The "power 2" has "undone" the sqrt operation.

Similarly, the "e to the power" operation undoes the "ln" expression in the example you are asking about.

Hope it makes sense.

## Re: Taking e to both sides

It does now. Thank you.

X

It does now.  Thank you.


# Law of Sines in Triangle

On the side BC of the triangle ABC we construct towards the exterior a square BCDE. Denote the intersection between AE and BC by M. Use the law of sines to prove that

$$\frac{BM}{CM}=\frac{\cos \measuredangle B\cdot \sin \measuredangle C}{\sqrt{2} \cdot \sin \measuredangle B \cdot \sin(\measuredangle C+45°)}$$

If someone could please help me prove this problem. I do not have a similar problem to work off of. I am unclear of where the $\sin(\measuredangle C+45^{\circ})$ comes into play and the $\sqrt{2}$.

• the end got cut off...i am unlcear where the sin(<C+45) and \sqrt{2} come into play. – user423388 Mar 8 '17 at 13:42
• Provide the figure – Nick Pavini Mar 8 '17 at 14:09

Let $\measuredangle B=\beta$, $\measuredangle C=\gamma$, $\measuredangle BAE=\alpha_1$, $\measuredangle CAE=\alpha_2$. By applying the sine rule to triangles $BAM$ and $CAM$ one gets: $${BM\over\sin\alpha_1}={AM\over\sin\beta},\quad {CM\over\sin\alpha_2}={AM\over\sin\gamma},\quad \hbox{whence:}\quad {BM\over CM}={\sin\alpha_1\over\sin\alpha_2}{\sin\gamma\over\sin\beta}.$$ By applying then the sine rule to triangles $BAE$ and $CAE$ one gets: $${\sin\alpha_1\over BE}={\sin(\beta+90°)\over AE},\quad {\sin\alpha_2\over CE}={\sin(\gamma+45°)\over AE}.$$ From that, taking into account that $CE=\sqrt2 BE$, one can readily obtain $\displaystyle{\sin\alpha_1\over\sin\alpha_2}$ and thus the desired result.

• fully understood this !! thank you so much ! – user423388 Mar 8 '17 at 16:34

Using sine rule at $ACE$ we have:

$$\frac{AE}{\sin(C+45°)}=\frac{AC}{\sin \alpha}$$

Using sine rule at $ABE$ we have:

$$\frac{AE}{\sin(B+90°)}=\frac{AB}{\sin \beta}$$

Dividing both equations we have:

$$\frac{\cos B}{\sin (C+45°)}=\frac{AC}{AB}\cdot \frac{\sin \beta}{\sin \alpha} \quad (1)$$

Using sine rule at $ABC$ we get

$$\frac{AC}{AB}=\frac{\sin B}{\sin C}$$

So, from $(1)$

$$\frac{\cos B\cdot \sin C}{\sin B\cdot \sin (C+45°)}=\frac{\sin \beta}{\sin \alpha}\quad (2)$$

but $\alpha + \beta=45°$ so

$$\frac{\sin \beta}{\sin \alpha}=\frac{\sin \beta}{\sin (45°-\beta)}=\sqrt{2}\cdot\frac{\tan \beta}{1-\tan \beta}\quad (3)$$

Finaly we can use, from the triangle BME, that

$$\tan \beta = \frac{BM}{BE}=\frac{BM}{BM+CM}\to \frac{BM}{CM}=\frac{\tan \beta}{1- \tan \beta} \quad (4)$$

Pluging $(3)$ and $(4)$ at $(2)$ we get what we want

• can you explain (3)? i dont understand how you switched from sine to tangent... – user423388 Mar 8 '17 at 16:17
• @erica: $\frac{\sin \beta}{\sin (45°-\beta)}=\frac{\sin \beta}{(\sqrt{2}/2)(\cos \beta-\sin \beta)}=$, now divide the numerator and denominator by $\cos \beta$ and get $\sqrt{2}\cdot\frac{\tan \beta}{1-\tan \beta}$ – Arnaldo Mar 8 '17 at 16:33

In triangle $ABM$, using the law of sines implies $$\frac{BM}{AM}=\frac{\sin\measuredangle BAM}{\sin\measuredangle B}$$ similarly, in $ACM$: $$\frac{CM}{AM}=\frac{\sin\measuredangle CAM}{\sin\measuredangle C}$$ combining these two yields: $$\frac{BM}{CM}=\frac{\sin\angle BAM}{\sin\angle CAM}\cdot\frac{\sin\angle C}{\sin\angle B}\label{*}\tag{*}$$ Now, in triangle $ABE$ note that $\measuredangle ABE=\measuredangle B+90^{\circ}$.

Thus $\sin\measuredangle ABE=\cos\measuredangle B$ and the law of sines implies $$\frac{\sin\measuredangle BAM}{\cos\measuredangle B}=\frac{BE}{AE}$$ and for triangle $ACE$: $$\frac{\sin\measuredangle CAM}{\sin(\measuredangle C+45)}=\frac{CE}{AE}$$ Note that $CE=\sqrt{2}BE$. Now combining the two equations implies: $$\frac{BE}{CE}=\frac 1{\sqrt 2}=\frac{\sin\measuredangle BAM}{\sin\measuredangle CAM}\cdot\frac{\sin(\measuredangle C+45)}{\cos\measuredangle B}$$ And finally, you can use $\eqref{*}$ to get the desired result.

• OOps. I am late to the party – polfosol Mar 8 '17 at 15:00
• can you explain how $\text{sin}\angleABE = \text{cos}\angleB$? – user423388 Mar 8 '17 at 15:58


## Unique Path I

A robot is located at the top-left corner of a m x n grid (marked 'Start' in the diagram below).
The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked 'Finish' in the diagram below).
How many possible unique paths are there?

## Analysis:

This is an easy problem. From the description we know that, the robot can only move down or right, which means, if the robot is now in position (x,y), then the position before this step must be  either (x-1,y) or (x, y-1). Since current position is only from these two previous positions, the number of possible paths that the robot can reach this current position (x,y) is the sum of paths from (x-1, y) and (x, y-1).

We want to get the number of paths on position (m,n), we need to know (m-1,n) and (m, n-1). For (m-1,n), we must know (m-2,n) and (m-1,n-1) ...  until we back to the start position (1,1) ([0][0] in  C++). Note that the boundary of the map, we can easily know that the top row and the first column of the map are all 1s. Use loop can solve the problem then.

### Code(C++):

class Solution {
public:
int uniquePaths(int m, int n) {
// Start typing your C/C++ solution below
// DO NOT write int main() function
int **arr = new int* [m];
for (int i=0;i<m;i++){
arr[i]= new int[n];
}
arr[0][0]=1;
for (int i=0;i<m;i++){
arr[i][0] = 1;
}
for (int i=0;i<n;i++){
arr[0][i] = 1;
}
for (int i=1;i<m;i++){
for(int j=1;j<n;j++){
arr[i][j] = arr[i][j-1] + arr[i-1][j];
}
}
return arr[m-1][n-1];
}
};

### Code(Python):

class Solution:
# @return an integer
def uniquePaths(self, m, n):
#define map and initialization
mp = [[0 for i in xrange(n)] for i in xrange(m)]
for i in xrange(m):
mp[i][0]=1
for j in xrange(n):
mp[0][j]=1

for i in xrange(1,m):
for j in xrange(1,n):
mp[i][j]=mp[i-1][j]+mp[i][j-1]

return mp[m-1][n-1]

1. I think most of your posts are great but sometimes it is very hard to understand your English though.

1. Actually I have been refining both the analysis and code in all my posts recently. Thank you very much for your suggestion.

2. Does this algorithm have a name?

3. Does this algorithm have a name?

1. Dynamic Programming

4. I LOVE YOU YU!

5. why did you initialise by 1 for the borders ?


# find length of rectangle given diagonal and area

• September 24th 2013, 11:08 AM
Bonganitedd
find length of rectangle given diagonal and area
Rectangle has area=168 m^2 and diagonal of 25. Find length
This is how tried to attempt the problem
Area= L X W
168 = L x W ..........(1)
L^2 + W^2 =25^2 ............(2)
From (1) L = 168/W...........(3)
Substitute (3) into (2)
(168/W)^2 +W^2 = 625
28224/W^2 + W^2 = 625
The problem gets complicated as I proceed
Is this aproach correct if it is,
Is there a convinient method
• September 24th 2013, 11:18 AM
votan
Re: find length of rectangle given diagonal and area
Quote:

Originally Posted by Bonganitedd
Rectangle has area=168 m^2 and diagonal of 25. Find length
This is how tried to attempt the problem
Area= L X W
168 = L x W ..........(1)
L^2 + W^2 =25^2 ............(2)
From (1) L = 168/W...........(3)
Substitute (3) into (2)
(168/W)^2 +W^2 = 625
28224/W^2 + W^2 = 625
The problem gets complicated as I proceed
Is this aproach correct if it is,
Is there a convinient method

Sketch a rectangle with one diagonal. laber the sides L and W and D for the diabonal. Write Pythagorean theorem, and the area = LW. Eliminate L from the the Pythagorean theorem and solve for W then find L from area formula.
• September 24th 2013, 11:30 AM
Bonganitedd
Re: find length of rectangle given diagonal and area
Its same approach I used, I did draw a rectangle now how do eliminate L bcos the pathygras is L^2 + W^2 =25^2
The area, LW =168
• September 24th 2013, 12:17 PM
Plato
Re: find length of rectangle given diagonal and area
Quote:

Originally Posted by Bonganitedd
Rectangle has area=168 m^2 and diagonal of 25. Find length
This is how tried to attempt the problem
Area= L X W
168 = L x W ..........(1)
L^2 + W^2 =25^2 ............(2)
From (1) L = 168/W...........(3)
Substitute (3) into (2)
(168/W)^2 +W^2 = 625
28224/W^2 + W^2 = 625
The problem gets complicated as I proceed
Is this aproach correct if it is,
Is there a convinient method

Have a look at this webpage.
• September 24th 2013, 01:15 PM
Soroban
Re: find length of rectangle given diagonal and area
Hello, Bonganitedd!

Quote:

Rectangle has area=168 m^2 and diagonal of 25. Find the length.

This is how tried to attempt the problem
$\text{Area} \:=\: L\cdot W \:=\:168 \quad\Rightarrow\quad L \,=\,\frac{168}{W}\;\;[1]$

$L^2 + W^2 \:=\:25^2\;\;[2]$

$\text{Substitute [1] into [2]: }\;\left(\frac{168}{W}\right)^2 +W^2 \:=\:625 \quad\Rightarrow\quad \frac{28,\!224}{W^2} + W^2 \:=\: 625$

Is this approach correct? . Yes
If it is, is there a convinient method?

We have: . $\frac{28,\!224}{W^2} + W^2 \:=\:625$

Multiply by $W^2\!:\;\;28,\!224 + W^4 \:=\:625W^2 \quad\Rightarrow\quad W^4 - 625W^2 + 28,\!224 \:=\:0$

Factor: . $(W^2 - 49)(W^2 - 576) \:=\:0$

$\begin{array}{ccccccccc}W^2-49 \:=\:0 & \Rightarrow & W^2 \:=\:49 & \Rightarrow & W \:=\:7 \\ W^2 - 576 \:=\:0 & \Rightarrow & W^2 \:=\:576 & \Rightarrow & W \:=\:24 \end{array}$

$\text{Assuming }L > W\text{, we have: }\:W\,=\,7,\;\boxed{L \,=\,24}$
• September 24th 2013, 02:08 PM
HallsofIvy
Re: find length of rectangle given diagonal and area
Once you get to "[tex]W^4- 625W^2+ 28224=0 as Soroban showed, if the "fourth degree polynomial" bothers you, you can let $x= W^2$ so that your equation is x^2- 625x+ 28224= 0 and use whatever method you like, completing the square or the quadratic formula, to solve that quadratic equation.
• September 24th 2013, 07:02 PM
Bonganitedd
Re: find length of rectangle given diagonal and area
Not taking away credit to other members who contributed, but @ Hallsofivy your advice is what I needed. Big u to MHF.


# Write $\cos^8(x)$ in terms of $\sin(x)$?

I am currently working on an integral using u substitution and have hit a wall at the above point. The integral in question is $\int$$\sin^7(x)$$\cos^9(x)$ using $u=sin(x)$.

After finding $du=cos(x) dx$, I am asked to express $\cos^8(x)$ in terms of $u$, at which point I am stuck. My current attempt was to reduce the power down from 8 to 2, and express it as $(1-u^2)$, but that gives me a wrong answer (this assignment checks for us before submission).

Any help is appreciated.

Thanks.

• What did you find the integral to be (your "wrong" answer)? – Henry Oct 18 '15 at 9:55
• I hadn't got to that point yet as the next few steps required it expressed in terms of u still. – Dwayne H Oct 18 '15 at 9:56
• OK, your "wrong" answer in terms of $u$? – Henry Oct 18 '15 at 9:57
• I'd only gotten to the point of $u^7(1-u^2)$, but seeing as the assignment said it was wrong I didn't try going ahead and solving for that. – Dwayne H Oct 18 '15 at 9:59
• $\displaystyle \int u^7(1-u^2)^4\,du$ with a power of $4$ seems more likely – Henry Oct 18 '15 at 10:00

If $$\cos^2 x = 1-\sin^2 x$$ then $$\cos^8 x = (1-\sin^2 x)^4.$$
Using $u=\sin x$, you have $du=\cos x\,dx$, and also $\cos^2x=1-u^2$, so $\cos^8x=(1-u^2)^4$. Thus your integral becomes $$\int u^7(1-u^2)^4\,du$$


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It is found that the effect of estimator variability is significant to obtain VaR forecast with better coverage. An investor is facing two potential financial losses and with the following joint density function: Let be the total of these two losses. Example: Plastic covers for CDs (Discrete joint pmf) Measurements for the length and width of a rectangular plastic covers for CDs are rounded to the nearest mm(so they are discrete). The correlation coefficient is equal to the covariance divided by the product of the standard deviations of the variables. More about Covariance. Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 2: Covariance and Correlation Section 5-4 Consider the joint probability distribution fXY(x;y). If instead, you want to get a step-by-step calculation of all descriptive statistics, you can try our descriptive statistics calculator. Let W have the exponential distribution with mean 1. Returns population covariance, the average of the products of deviations for each data point pair in two data sets. Expected Return Formula Calculator. If A is a scalar, cov(A) returns 0. Covariance • Variance and Covariance are a measure of the "spread" of a set of points around their center of mass (mean) • Variance - measure of the deviation from the mean for points in one dimension e. Covariance is driving me nuts, is there any simple way to make the calculator do the work for you? Get your mind off your Level I results with a free 2020 Level II JumpStart package. 35 5% -3% Given The Above Information On Two Investments A And B, Calculate The Statistics Below. Please enter the necessary parameter values, and then click 'Calculate'. Suppose, under uncertainty, the manager believes that a risky asset, for example, an equity, can bring him different results, which at the moment of portfolio formation can only be judged with some probability, as shown in Table. A low value means there is a weak relationship. CSC 411 / CSC D11 / CSC C11 Probability Density Functions (PDFs) The off-diagonal terms are covariances: Σ ij = cov(x i,x j) = E p(x)[(x i −µ i)(x j −µ j)] (10) between variables x i and x j. • Probability and Statistics for Engineering and the Sciences, Covariance, Correlation, Sampling Distributions, Central Limit calculator or for any other. E(X1)=µX1 E(X2)=µX2 var(X1)=σ2 X1 var(X2)=σ2 X2 Also, we assume that σ2 X1 and σ2 X2 are ﬁnite positive values. Estimating the uncertainty of revenues and investment decisions. And, to calculate the probability of an interval, you take the integral of the probability density function over it. Byju's Covariance Calculator is a tool which makes calculations very simple and interesting. The covariance matrix is a matrix that only concerns the relationships between variables, so it will be a k x k square matrix. Using the formulae above to compute covariance can sometimes be tricky. The covariance matrix can be calculated in NumPy using the cov() function. The sign (+ or -) of the correlation affects its interpretation. I can of course go ahead and derive all the elements of conditional covariance matrix. The Lognormal Probability Distribution Let s be a normally-distributed random variable with mean µ and σ2. The denominator is represented by (n-1), which is just one less than the number of data pairs in your data set. The converse. There is a 50 percent probability that after one year the stock's price will rise to $5 (yielding a 400 percent return). R Functions for Probability Distributions. The probability density function of the bivariate normal distribution is implemented as MultinormalDistribution[mu1, mu2, sigma11, sigma12, sigma12, sigma22] in the Wolfram Language package MultivariateStatistics. 128 CHAPTER 7. Variables are inversely. If the variables tend to show similar behavior, the covariance is positive. Uniform distribution Calculator - High accuracy calculation Welcome, Guest. This post provides practice problems to reinforce the concepts discussed in the preceding post on univariate probability distributions. That is, covariance matrices with small determinants denote variables that are redundant or highly correlated. Variables are positively related if they move in the same direction. calculate and interpret covariance and correlation and interpret a scatterplot; calculate and interpret the expected value, variance, and standard deviation of a random variable and of returns on a portfolio; calculate and interpret covariance given a joint probability function; calculate and interpret an updated probability using Bayes' formula;. This online calculator computes covariance between two discrete random variables. By default, this function will calculate the sample covariance matrix. The sign (+ or -) of the correlation affects its interpretation. 2 Covariance Covariance is a measure of how much two random variables vary together. Chapter 4 Variances and covariances Page 3 A pair of random variables X and Y is said to be uncorrelated if cov. 1 Introduction. Joyce, Fall 2014 Covariance. [In our case, a 5×5 matrix. Also, it can be considered as a generalization of the concept of variance of two random variables. Covariance Calculator calculator, formula and. The Covariance Calculator an online tool which shows Covariance for the given input. Covariance gives you a positive number if the variables are positively related. Covariance[dist] gives the covariance matrix for the multivariate symbolic distribution dist. The formula defines covariance for discrete variables in Simon & Blume (1994): Mathematics for Economists, end of section A5. Linear Models in SAS (Regression & Analysis of Variance) The main workhorse for regression is proc reg, and for (balanced) analysis of variance, proc anova. In case the greater values of one variable are linked to the greater values of the second variable considered, and the same corresponds for the smaller figures, then the covariance is positive and is a signal that the two variables show similar behavior. If the variables tend to show similar behavior, the covariance is positive. Obtaining covariance estimates between variables allows one to better estimate direct and indirect effects with other variables, particularly in complex models with many parameters to be estimated. If using percent form, the user must add the percent sign (%) at the end of the number. Covariance[v1, v2] gives the covariance between the vectors v1 and v2. In the matrix diagonal there are variances, i. (2005), Fundamentals of Probability with Stochastic Processes, Roussas, G. In addition, we may only be able to assess unconditional coverage probability for VaR forecast of the SVAR model. Problem 31B. I have a joint probability mass function of two variables X,Y like here. P function in Microsoft Excel. Joint Discrete Probability Distributions. Chapter 4 Variances and covariances Page 3 A pair of random variables X and Y is said to be uncorrelated if cov. I hope you found this video useful, please subscribe for daily videos! WBM Foundations: Mathematical logic Set theory Algebra: Number theory Group theory Lie groups Commutative rings Associative. By using this formula, after calculation, you can verify the result of such calculations by using our covariance calculator. Covariance is a measure of how much two random variables vary together. When comparing data samples from different populations, two of the most popular measures of association are covariance and correlation. Applied to historical prices, covariance can help determine if stocks' prices tend. Video for finding the covariance and correlation coefficient by hand. TI-82 / TI-83 Graphing Calculator. Correlation values range from positive 1 to negative 1. 5 should display. For a discrete variable$\sum_{i=1}^nP(X=x_{i}|A)=1$, hence we immediately can fill in the missing values in the conditional distribution tables. The TI-83 Graphing Calculator can facilitate the entry of ordered lists of data and perform some statistical analyses, but lacks a single command to calculate the covariance of two lists of numbers. Covariance & Correlation The covariance between two variables is defined by: cov x,y = x x y y Can I directly relate the free parameters to the covariance matrix? First calculate P(x) by marginalizing over y: P xˆexp{1 21 2 x x0 x 2} dyexp{1 21 2 [y y0 y 2 2 x x0 x y y0 y]} probability content is contained within a radius of s(2. Lecture 21: Conditional Distributions and Covariance / Correlation Statistics 104 Colin Rundel April 9, 2012 6. when the returns of one asset goes up, the. 92 and 202-205; Whittaker and Robinson 1967, p. For instance, I have been given a discrete random variable X with probability function px(x) = 1/2 if x = -1, 1/4 if x = 0, 1/4 if x = 1, 0 otherwise. Discrete Random Variable Calculator. Now, we also need to be able to calculate the covariance and correlation for a joint probability function. It is considered the ideal case in which the probability structure underlying the categories is known perfectly. Although the covariance and variance are linked to each other in the above manner, their probability distributions are not attached to each other in a simple manner and have to be dealt separately. Descriptive Statistics which contains one variable and multivariable calculators for 20 descriptive statistics measures including: mean, variance, covariance, quantile, interquartile range, correlation and many more. There are various kinds of insurance. Recall that , and that is the normal density with mean and variance. In case the greater values of one variable are linked to the greater values of the second variable considered, and the same corresponds for the smaller figures, then the covariance is positive and is a signal that the two variables show similar behavior. Video and lecture notes from a tutorial on Probability and Statistics given at PyData NYC 2019. For each of the three factors, the probability is 0. when the returns of one asset goes up, the. This calculator is featured to generate the complete work with steps for any corresponding input values may helpful for grade school students to solve covaraince worksheet or homework problems or. The expected value of X is usually. Imagine how confusing it would be if people used degrees Celsius and degrees Fahrenheit interchangeably. 2 thoughts on " An Example on Calculating Covariance. Probability helps the company to calculate how many chances that insurance holders have to claim the insurance. it helps us to understand how two sets of data are related to each other. For example, if the number of desired outcomes divided by the number of possible events is. Use the theorem we just proved to calculate the covariance of X and Y. Sample Mean and Covariance Calculator. My ultimate objective is to calculate with a given set data the following ratio: Det[Covariance matrix for X]/Det[Conditional covariance matrix X|S] Any idea about how to achieve this?. Most articles and reading material on probability and statistics presume a basic understanding of terms like means, standard deviation, correlations, sample sizes and covariance. CSC 411 / CSC D11 / CSC C11 Probability Density Functions (PDFs) The off-diagonal terms are covariances: Σ ij = cov(x i,x j) = E p(x)[(x i −µ i)(x j −µ j)] (10) between variables x i and x j. - The probability that a company will pass the test given that it will subsequently survive 12 months, is. In the next section, read Problem 1 for an example showing how to turn raw data into a variance-covariance matrix. Covariance Calculator calculator, formula and work with steps to estimate the relationship (linear dependence) between two dataset in statistical experiments. WORKED EXAMPLES 3 COVARIANCE CALCULATIONS EXAMPLE 1 Let Xand Y be discrete random variables with joint mass function defined by f X,Y(x,y) = 1 4 Hence the two variables have covariance and correlation zero. I know the definition of covariance and I'm trying to solve some exercises. Rules for the Correlation Coefficient. Standard Deviation Calculator Variance Calculator Kurtosis Calculator Skewness Calculator. Using the formulae above to compute covariance can sometimes be tricky. com's Covariance calculator is an online statistics & probability tool to estimate the nature of association between two random variables X & Y in probability & statistics experiments. Before we demonstrate the function in Excel, a few observations on the covariance and correlation measures. The probability that a large earthquake will occur on the San Andreas Fault in. If you see any typos, potential edits or changes in this Chapter, please note them here. function [probability] = comp_gauss_dens_val (mean, covarianceMatrix, givenMatrix ) The value assigned in lengthofM varibale is the length of the mean of the given matrix, which in our is one dimensional in our case. If an input is given then it can easily show the result for the given number. Joint Discrete Probability Distributions. A negative covariance means that the variables are inversely related, or that they move in opposite directions. If A is a vector of observations, C is the scalar-valued variance. All you can really tell from covariance is if there is a positive or negative relationship. Apart from the six-page limitation, originality, quality and clarity will be the criteria for choosing the material to be published in Statistics & Probability Letters. Sample covariance measures the …. The domain of t is a set, T , of real numbers. Provide an interpretation (10%). Covariance[m] gives the covariance matrix for the matrix m. Formulas that calculate covariance can predict how two stocks might perform relative to each other in the future. Click the Calculate! button and find out the covariance matrix of a multivariate sample. (i) The expected value measures the center of the probability distribution - center of mass. covariance calculator - step by step calculation to measure the statistical relationship (linear dependence) between the two sets of population data, along with. For example,. The correlation coefficient is also known as the Pearson Product-Moment Correlation Coefficient. pX Beta calculator. calculate and interpret covariance and correlation and interpret a scatterplot; calculate and interpret the expected value, variance, and standard deviation of a random variable and of returns on a portfolio; calculate and interpret covariance given a joint probability function; calculate and interpret an updated probability using Bayes' formula;. In probability, we use 0. Adding a constant to a random variable does not change their correlation coefficient. is between and inclusive, which meets the first property of the probability distribution. ) S is said to have a lognormal distribution,. Full curriculum of exercises and videos. Small sample size problems. Different categories of descriptive measures are introduced and discussed along with the Excel functions to calculate them. In introductory finance courses, we are taught to calculate the standard deviation of the portfolio as a measure of risk, but part of this calculation is the covariance of these two, or more, stocks. To find the covariance matrix you need to calculate the Fisher information matrix. A covariance refers to the measure of how two random variables will change together and is used to calculate the correlation between variables. Obtaining covariance estimates between variables allows one to better estimate direct and indirect effects with other variables, particularly in complex models with many parameters to be estimated. In introductory finance courses, we are taught to calculate the standard deviation of the portfolio as a measure of risk, but part of this calculation is the covariance of these two, or more, stocks. Do October 10, 2008 A vector-valued random variable X = X1 ··· Xn T is said to have a multivariate normal (or Gaussian) distribution with mean µ ∈ Rn and covariance matrix Σ ∈ Sn 1. covariance calculator - step by step calculation to measure the statistical relationship (linear dependence) between the two sets of population data, along with. This calculator determines the following coin toss probability scenarios * Coin Toss Sequence such as HTHHT * Probability of x heads and y tails * Covariance of X and Y denoted Cov(X,Y) * The. Is there a relationship between Xand Y? If so, what kind? If you’re given information on X, does it give you information on the distribution of Y? (Think of a conditional distribution). It's an online statistics and probability tool requires two sets of population data X and Y and measures of how much these data sets vary together, i. When the covariance is positive, X tends to be high when Y is high, and vice versa; when the covariance is negative, X tends to be high when Y is low, and vice versa. Consider two random variables$X$and$Y$. Here are some documents to help you use the TI-82 calculator. Enter X values (Separated by comma) Enter Y values (Separated by comma) Letter Arrangment Probability Calculator. Where Cov (A, B) – is covariance of portfolios A and B. Calculate the standard deviation of the returns using STDEV function; Finally, we calculate the VaR for 90, 95, and 99 confidence level using NORM. Or are they. The converse. 01 respectively for the VaR(90), VaR(95), and VaR(99). If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values, (i. S function is:. , the covariance of each element with itself. This suggests the question: Given a symmetric, positive semi-de nite matrix, is it the covariance matrix of some random vector?. The calculator will find the p-value for two-tailed, right-tailed and left-tailed tests from normal, Student's (T-distribution), chi-squared and Fisher (F-distribution) distributions. How to Calculate Expected Value. The probability that a large earthquake will occur on the San Andreas Fault in. After cleaning the data, the researcher must test the assumptions of ANOVA. Therefore, it is a straightforward exercise to calculate the correlation between X and. 2 Covariance Covariance is a measure of how much two random variables vary together. More about Covariance. Welcome to StatCalculators. The sample variance, s², is used to calculate how varied a sample is. This calculator computes the variance from a data set: To calculate the variance from a set of values, specify whether the data is for an entire population or from a sample. Other JavaScript in this series are categorized under different areas of applications in the MENU section on this page. It is based on the probability-weighted average of the cross-products of the random variables’ deviations from their expected values for each possible outcome. (b) In contrast to the expectation, the variance is not a linear operator. Covariance is a statistical calculation that helps you understand how two sets of data are related to each other. And, we are given that the standard deviation of X is 1/2, and the standard deviation of Y is the square root of 1/2. Correlation between the two variables is a normalized version of the Covariance. In probability theory and statistics, covariance is a measure of the joint variability of two random variables. Analysis of covariance (ANCOVA) allows to compare one variable in 2 or more groups taking into account (or to correct for) variability of other variables, called covariates. described with a joint probability mass function. rameterized by a mean vector µ, and a variance-covariance matrix Σ, written X ∼ (µ,Σ). Given such information it is possible to calculate the marginal distributions and then the joint distribution follows. The function underlying its probability distribution is called a probability density function. Standard Deviation Calculator Variance Calculator Kurtosis Calculator Skewness Calculator. A zero covariance may indicate that the two assets are independent. This continues our exploration of the semantics of the inner product. CSC 411 / CSC D11 / CSC C11 Probability Density Functions (PDFs) The off-diagonal terms are covariances: Σ ij = cov(x i,x j) = E p(x)[(x i −µ i)(x j −µ j)] (10) between variables x i and x j. search(“distribution”). The Multivariate Gaussian Distribution Chuong B. Probability is the likelihood of something happening or being true. Both of these two determine the relationship and measures the dependency between two random. The covariance of two variables tells you how likely they are to increase or decrease simultaneously. 128 CHAPTER 7. Then m is the vector of means and V is the variance-covariance matrix. Online probability calculator to find expected value E(x), variance (σ 2) and standard deviation (σ) of discrete random variable from number of outcomes. Covariance and correlation show that variables can have a positive relationship, a negative relationship, or no relationship at all. Covariance correlations in collision avoidance probability calculations. Printer-friendly version Introduction. Begin your Level II studies with a FREE Schweser JumpStart Package. If you're behind a web filter, please make sure that the domains *. The TI-83 Graphing Calculator can facilitate the entry of ordered lists of data and perform some statistical analyses, but lacks a single command to calculate the covariance of two lists of numbers. The calculator based methods proposed above don’t account for the fact that a probability is associated with asset returns. Doing the calculation To calculate the beta coefficient for a we'll compare how the stock and the index move relative to each other with a covariance formula and then divide that result by the. 329) and is the covariance. ) q for "quantile", the inverse c. This calculator is featured to generate the complete work with steps for any corresponding input values may helpful for grade school students to solve. When a portfolio includes two risky assets, the Analyst needs to take into account expected returns, variances and the covariance (or correlation) between the assets' returns. These usual, our starting point is a random experiment with a probability measure ℙ on an underlying sample space. Bayesian decision theory is a fundamental statistical approach to the problem of pattern classification. Correlation values range from positive 1 to negative 1. Returns population covariance, the average of the products of deviations for each data point pair in two data sets. Variance and Standard Deviation of a Random Variable. Hey Flashcop and welcome to the forums. That is, if one increases, the other increases. It's an online statistics and probability tool requires two sets of population data X and Y` and measures of how much these data sets vary together, i. If an input is given then it can easily show the result for the given number. In this section, we will study an expected value that measures a special type of relationship between two real-valued variables. The Covariance Calculator an online tool which shows Covariance for the given input. Input the matrix in the text field below in the same format as matrices given in the examples. I was concurrently taking a basic theoretical probability and statistics, so even the idea of variance was still vague to me. 4, and in Robert J. Applied to historical prices, covariance can help determine if stocks' prices tend. Descriptive Statistics which contains one variable and multivariable calculators for 20 descriptive statistics measures including: mean, variance, covariance, quantile, interquartile range, correlation and many more. J Matney a Collision-avoidance calculations between orbiting objects make use of covariance matrices to characterize the uncertainty of the orbital position in space and time. Byju's Covariance Calculator is a tool which makes calculations very simple and interesting. The Multivariate Gaussian Distribution Chuong B. We will go through a review of probability concepts over here, all of the review materials have been adapted from CS229 Probability Notes. A portfolio is the total collection of all investments held by an individual or institution, including stocks, bonds, real estate, options, futures, and alternative investments, such as gold or limited partnerships. 2 Covariance Covariance is a measure of how much two random variables vary together. 25, multiply the answer by 100 to get 25%. The uniform distribution is used to describe a situation where all possible outcomes of a random experiment are equally likely to occur. is the correlation of and (Kenney and Keeping 1951, pp. 12 that she has exactly these two risk factors (but not the other). In general, if there are n random variables, the outcome is an n-dimensional vector of them. Stock Correlation Calculator. All we'll be doing here is getting a handle on what we can expect of the correlation coefficient if X and Y are independent, and what we can expect of the correlation coefficient if X and Y are dependent. If the probability of the event changes when we take the first event into consideration, we can safely say that the probability of event B is dependent of the occurrence of event A. As you doubtless know, the variance of a set of numbers is defined as the "mean squared difference from the mean". This online calculator computes covariance between two discrete random variables. Let X 1 = number of dots on the red die X 2 = number of dots on the green die. The correlation coefficient is equal to the covariance divided by the product of the standard deviations of the variables. How to Calculate Covariance. Let's now look at how to calculate the risk of the portfolio. As a simple example of covariance we'll return once again to the Old English example of Section 2. This tool provides direct calculations for a variety of probability distributions. 5 Covariance and Correlation Covariance and correlation are two measures of the strength of a relationship be-tween two r. ] Before constructing the covariance matrix, it’s helpful to think of the data matrix as a collection of 5 vectors, which is how I built our data matrix in R. We construct a non-separable space-time covariance function based on a diffusive Langevin equation. If the two variables are dependent then the covariance can be measured using the following formula:. Although the covariance and variance are linked to each other in the above manner, their probability distributions are not attached to each other in a simple manner and have to be dealt separately. A positive covariance would indicate a positive linear relationship between the variables, and a negative covariance would indicate the opposite. The syntax of the Covariance. It is actually used for computing the covariance in between every column of data matrix. It is also helpful for insurance. They must then calculate the F-ratio and the associated probability value (p-value). By using this formula, after calculation, you can verify the result of such calculations by using our covariance calculator. If A is a matrix whose columns represent random variables and whose rows represent observations, C is the covariance matrix with the corresponding column variances along the diagonal. [In our case, a 5×5 matrix. It just creates confusion because they are not equivalent. In order to calculate the Student T Value for any degrees of freedom and given probability. Key Differences Between Covariance and Correlation. With the covariance option, correlate can be used to obtain covariance matrices, as well as correlation matrices, for both weighted and unweighted data. It's the statistics & probability functions formula reference sheet contains most of the important functions for data analysis. In my first machine learning class, in order to learn about the theory behind PCA (Principal Component Analysis), we had to learn about variance-covariance matrix. A distribution is described as normal if there is a high probability that any observation form the population sample will have a value that is close to the mean, and a low probability of having a value that is far from the mean. So, I wrote an add-in that used the matrix algebra functions to create a covariance matrix that would change if you changed the data. 05 Jeremy Orlo and Jonathan Bloom 1 Learning Goals 1. By default, this function will calculate the sample covariance matrix. Calculate the covariance of and. The function is new in Excel 2010 and so is not available in earlier versions of Excel. How does this covariance calculator work? In data analysis and statistics, covariance indicates how much two random variables change together. A Statistics Co-Variance Calculator An online Co-Variance Calculator to measure of two variables X and Y. Covariance and Correlation Deﬁnition: Covariance Let X and Y be two RV’s with means x and y, respectively. In statistical theory, covariance is a measure of how much two random variables change together. E(X1)=µX1 E(X2)=µX2 var(X1)=σ2 X1 var(X2)=σ2 X2 Also, we assume that σ2 X1 and σ2 X2 are ﬁnite positive values. 2 Covariance Covariance is a measure of how much two random variables vary together. Expectation of a Function of a Random Variable Suppose that X is a discrete random variable with sample space Ω, and φ(x) is a real-valued function with domain Ω. A sample is a randomly chosen selection of elements from an underlying population. Analysis of covariance (ANCOVA) allows to compare one variable in 2 or more groups taking into account (or to correct for) variability of other variables, called covariates. It is important in security analysis to determine how much or how little price movements in two companies or industries are connected. These topics are somewhat specialized, but are particularly important in multivariate statistical models and for the multivariate normal distribution. Using the covariance formula, you can determine whether economic growth and S&P 500 returns have a positive or inverse relationship. Typically, the population is very large, making a complete enumeration of all the values in the population impossible. How does this covariance calculator work? In data analysis and statistics, covariance indicates how much two random variables change together. Consider two random variables$X$and$Y\$. Understand the meaning of covariance and correlation. If is the covariance matrix of a random vector, then for any constant vector ~awe have ~aT ~a 0: That is, satis es the property of being a positive semi-de nite matrix. The problem is solved by standardize the value of covariance (divide it by ˙. An investor is facing two potential financial losses and with the following joint density function: Let be the total of these two losses. It's the statistics & probability functions formula reference sheet contains most of the important functions for data analysis. S function is:. Population Standard Deviation The population standard deviation, the standard definition of σ , is used when an entire population can be measured, and is the square root of the variance of a given data set. Excel does such a great job in calculating correlation and covariance that it is not necessary to memorize the formulas of covariance and correlation, but here they are, along with examples worked out in Excel: Covariance of variables x and y from a known population = σ xy. The covariance for two random variates X and Y, each with sample size N, is defined by the expectation…. Calculate the covariance of and. Discrete Random Variable Calculator. Properties of variance and covariance (a) If and are independent, then by observing that. A portfolio is the total collection of all investments held by an individual or institution, including stocks, bonds, real estate, options, futures, and alternative investments, such as gold or limited partnerships. On this page, we'll begin our investigation of what the correlation coefficient tells us. Consider the following example: Example. The covariance is defined as. Click the Calculate! button and find out the covariance matrix of a multivariate sample. A randomly selected day was a long commute. These usual, our starting point is a random experiment with a probability measure ℙ on an underlying sample space. For example, the probability of a two-dimensional case, in which the vector of random variables is X = [X, Y] T, can be calculated as. This matrix is not usually printed or saved. , the variables tend to show similar behavior), the covariance is positive. For instance, I have been given a discrete random variable X with probability function px(x) = 1/2 if x = -1, 1/4 if x = 0, 1/4 if x = 1, 0 otherwise. If the covariance is a large positive number, then we expect x i to be largerthanµ iwhenx j islargerthanµ j. Let's calculate the covariance for the example age and income data. In probability theory and statistics, covariance is a measure of the joint variability of two random variables. Covariance and correlation show that variables can have a positive relationship, a negative relationship, or no relationship at all. You COULD do a calculation patterned after covariance with 3 or more variables, but I don’t see it as meaning anything, and I don’t think it would be admissable as a valid statistical function. (The covariance of X and Y is sometimes written cov(X, Y). He also covers testing hypotheses, modeling different data distributions, and calculating the covariance and correlation between data sets.


# Kendall tau distance

The Kendall tau rank distance is a metric that counts the number of pairwise disagreements between two ranking lists. The larger the distance, the more dissimilar the two lists are. Kendall tau distance is also called bubble-sort distance since it is equivalent to the number of swaps that the bubble sort algorithm would take to place one list in the same order as the other list. The Kendall tau distance was created by Maurice Kendall.

## Definition

The Kendall tau ranking distance between two lists $\displaystyle{ \tau_1 }$ and $\displaystyle{ \tau_2 }$ is

$\displaystyle{ K(\tau_1, \tau_2) = |\{(i,j): i \lt j, ( \tau_1(i) \lt \tau_1(j) \wedge \tau_2(i) \gt \tau_2(j) ) \vee ( \tau_1(i) \gt \tau_1(j) \wedge \tau_2(i) \lt \tau_2(j) )\}|. }$

where

• $\displaystyle{ \tau_1(i) }$ and $\displaystyle{ \tau_2(i) }$ are the rankings of the element $\displaystyle{ i }$ in $\displaystyle{ \tau_1 }$ and $\displaystyle{ \tau_2 }$ respectively.

$\displaystyle{ K(\tau_1,\tau_2) }$ will be equal to 1 if the two lists are identical and $\displaystyle{ -1 }$ (where $\displaystyle{ n }$ is the list size) if one list is the reverse of the other. The normalized Kendall tau distance therefore lies in the interval [-1,1].

Kendall tau distance may also be defined as

$\displaystyle{ K(\tau_1,\tau_2) = \begin{matrix} \sum_{\{i,j\}\in P} \bar{K}_{i,j}(\tau_1,\tau_2) \end{matrix} }$

where

• P is the set of unordered pairs of distinct elements in $\displaystyle{ \tau_1 }$ and $\displaystyle{ \tau_2 }$
• $\displaystyle{ \bar{K}_{i,j}(\tau_1,\tau_2) }$ = 0 if i and j are in the same order in $\displaystyle{ \tau_1 }$ and $\displaystyle{ \tau_2 }$
• $\displaystyle{ \bar{K}_{i,j}(\tau_1,\tau_2) }$ = 1 if i and j are in the opposite order in $\displaystyle{ \tau_1 }$ and $\displaystyle{ \tau_2. }$

Kendall tau distance can also be defined as the total number of discordant pairs.

Kendall tau distance in Rankings: A permutation (or ranking) is an array of N integers where each of the integers between 0 and N-1 appears exactly once. The Kendall tau distance between two rankings is the number of pairs that are in different order in the two rankings. For example, the Kendall tau distance between 0 3 1 6 2 5 4 and 1 0 3 6 4 2 5 is four because the pairs 0-1, 3-1, 2-4, 5-4 are in different order in the two rankings, but all other pairs are in the same order.[1]

If Kendall tau function is performed as $\displaystyle{ K(L1,L2) }$ instead of $\displaystyle{ K(\tau_1,\tau_2) }$ (where $\displaystyle{ \tau_1 }$ and $\displaystyle{ \tau_2 }$ are the rankings of $\displaystyle{ L1 }$ and $\displaystyle{ L2 }$ elements respectively), then triangular inequality is not guaranteed. The triangular inequality fails in cases where there are repetitions in the lists. So then we are not dealing with a metric anymore.

## Example

Suppose one ranks a group of five people by height and by weight:

Person A B C D E
Rank by height 1 2 3 4 5
Rank by weight 3 4 1 2 5

Here person A is tallest and third-heaviest, and so on.

In order to calculate the Kendall tau distance, pair each person with every other person and count the number of times the values in list 1 are in the opposite order of the values in list 2.

Pair Height Weight Count
(A,B) 1 < 2 3 < 4
(A,C) 1 < 3 3 > 1 X
(A,D) 1 < 4 3 > 2 X
(A,E) 1 < 5 3 < 5
(B,C) 2 < 3 4 > 1 X
(B,D) 2 < 4 4 > 2 X
(B,E) 2 < 5 4 < 5
(C,D) 3 < 4 1 < 2
(C,E) 3 < 5 1 < 5
(D,E) 4 < 5 2 < 5

Since there are four pairs whose values are in opposite order, the Kendall tau distance is 4. The normalized Kendall tau distance is

$\displaystyle{ \frac{4}{5(5 - 1)/2} = 0.4. }$

A value of 0.4 indicates that 40% of pairs differ in ordering between the two lists.

## Computing the Kendall tau distance

A naive implementation in Python (using NumPy) is:

import numpy as np

def normalised_kendall_tau_distance(values1, values2):
"""Compute the Kendall tau distance."""
n = len(values1)
assert len(values2) == n, "Both lists have to be of equal length"
i, j = np.meshgrid(np.arange(n), np.arange(n))
a = np.argsort(values1)
b = np.argsort(values2)
ndisordered = np.logical_or(np.logical_and(a[i] < a[j], b[i] > b[j]), np.logical_and(a[i] > a[j], b[i] < b[j])).sum()
return ndisordered / (n * (n - 1))

However, this requires $\displaystyle{ n^2 }$ memory, which is inefficient for large arrays.

Given two rankings $\displaystyle{ \tau_1,\tau_2 }$, it is possible to rename the items such that $\displaystyle{ \tau_1 = (1,2,3,...) }$. Then, the problem of computing the Kendall tau distance reduces to computing the number of inversions in $\displaystyle{ \tau_2 }$—the number of index pairs $\displaystyle{ i,j }$ such that $\displaystyle{ i\lt j }$ while $\displaystyle{ \tau_2(i) \gt \tau_2(j) }$. There are several algorithms for calculating this number.

• A simple algorithm based on merge sort requires time $\displaystyle{ O(n \log n) }$.[2]
• A more advanced algorithm requires time $\displaystyle{ O(n\sqrt{\log{n}}) }$.[3]

Here is a basic C implementation.

#include <stdbool.h>

int kendallTau(short x[], short y[], int len) {
int i, j, v = 0;
bool a, b;

for (i = 0; i < len; i++) {
for (j = i + 1; j < len; j++) {

a = x[i] < x[j] && y[i] > y[j];
b = x[i] > x[j] && y[i] < y[j];

if (a || b)
v++;

}
}

return abs(v);
}

float normalize(int kt, int len) {
return kt / (len * (len - 1) / 2.0);
}


# True or False : If $f(x)$ and $f^{-1}(x)$ intersect at an even number of points , all points lie on $y=x$

Previously I have discussed about odd number of intersect points (See : If the graphs of $f(x)$ and $f^{-1}(x)$ intersect at an odd number of points, is at least one point on the line $y=x$?) Now , I want to know the even condition . For example $f(x) = \sqrt{x}$ and $f^{-1}(x) = x^2 , x\ge 0$ intersects each other in $(0,0)$ and $(1,1)$ points and these points located on $y=x$ line

Edit : Consider $f$ is continuous function.

• How would any intersection point not lie on $y=x$? The graphs of the function and its inverse (if it exists) is symmetric about $y=x$, so any intersection must be on $y=x$ Feb 23 '17 at 15:58
• @user160738 No , for example consider $f(x) = \frac{1}{x}$ and $f^{-1}(x) = \frac{1}{x}$ Feb 23 '17 at 16:00
• @user160738 another example is $f(x) = -x^3$ and $f^{-1}(x) = -\sqrt[3]{x}$ . They intersect each other in $(1 , -1)$ that is not on $y=x$ line. See : math.stackexchange.com/questions/1452098/… Feb 23 '17 at 16:08

The continuous case:

If the domain is not an interval, we can still find a counterexample. Define $$f(x)=\cases{2x+2 & if x\in(0,2),\cr x-3 & if x\in(3,5),}$$ then $$f^{-1}(x)=\cases{x/2-1 & if x\in(2,6),\cr x+3 & if x\in(0,2),}$$ and intersections are $(1,4)$, $(4,1)$.

However, if the domain is connected, it turns out to be true.

Proof of the theorem in the case of a connected domain (interval)

For contradiction, let us suppose that there is an even number of intersections and an intersection is out of diagonal.

1. We assume that $f^{-1}$ exists which means that $f$ is injective. Moreover, $f$ is continuous and its domain is interval, so $f$ is increasing or decreasing.
2. $f$ and $f^{-1}$ are symmetric by diagonal, therefore the set of intersections is symmetric by the diagonal.
3. Let denote $(x_1, x_2)$ an intersection out of diagonal. By symmetry, there is an intersection $(x_2, x_1)$. We can therefore WLOG assume that $x_1<x_2$.
4. $f(x_1) > x_1$ and $f(x_2) < x_2$, so by continuity, $f$ intersects the diagonal.
5. There is an even number of intersections and by symmetry, there is an even number of intersections out of diagonal. So there is even number of intersections on the diagonal. So there are at least two of them: $f(x_3) = x_3$, $f(x_4) = x_4$
6. $x_1<x_2$ and $f(x_1) > f(x_2)$, so $f$ is decreasing (on the whole, by 1).
7. $x_3<x_4$ and $f(x_3) < f(x_4)$, so $f$ is increasing.

• Okay but it's true in many cases in connected domain . Can you provide an example that it's wrong in connected domain ? Mar 3 '17 at 10:45
• Okay so why it's true for many examples ? Mar 3 '17 at 15:14
• You proved the theorem is wrong .okay , now provide some examples that shows theorem is false . Mar 3 '17 at 15:38
• Oh , I'm really sorry let me to read again . Mar 3 '17 at 15:40
• Sorry, I don't understand step 4 in your purported proof. Mar 4 '17 at 10:45

Assume that $f$ is a continuous and invertible real function with a connected domain. Then $f$ is either strictly decreasing or strictly increasing over its domain (or it would assume some value twice, and would not be one-to-one). Consider these two cases:

1. $f$ is decreasing. Then, since $f(x)-x$ is also decreasing, $f$ can't cross $y=x$ more than once. If it never crosses $y=x$, then it lies entirely above or below that line, and its inverse lies entirely on the other side; they never meet, and the theorem holds vacuously. If, on the other hand, it crosses $y=x$ exactly once, then the total number of intersections between $f$ and $f^{-1}$ is odd, since off-diagonal intersections come in pairs. The theorem holds in this case too.

2. $f$ is increasing. Then it cannot intersect $f^{-1}$ at any point off the line $y=x$. Suppose it did, at a pair of points $(x,y)$ and $(y,x)$ with $x<y$: then we would have $f(x)=y > x=f(y)$, contradicting the fact that $f$ is increasing. In this case, then, all intersections are on $y=x$, and the theorem holds.

Since the theorem holds whether $f$ is increasing or decreasing, it is true in general.

• For all decreasing functions we can say $f(x) - x$ is also decreasing ? Is it right ? Mar 9 '17 at 6:03
• Sure. A decreasing function minus an increasing function decreases even faster. Mar 9 '17 at 20:48
• Okay and how you can conclude that it intersects at most one time ? Mar 9 '17 at 20:55

Your question, as it currently stands definitely does not hold, as you do not specify the domain, continuity or conditions on inverse. This answer does define an $f$ on the whole real line that has a proper inverse.

Consider the function: \begin{align} f(x) = \begin{cases} 1 &\text{if x = 0}\\ 0 &\text{if x = 1}\\ x^2 &\text{if x > 0 and x\neq 1}\\ -2x &\text{if x < 0} \end{cases} \end{align} which has inverse \begin{align} f^{-1}(y) = \begin{cases} 1 &\text{if y = 0}\\ 0 &\text{if y = 1}\\ \sqrt{x} &\text{if x > 0 and x\neq 1}\\ -\tfrac{x}{2} &\text{if x < 0} \end{cases} \end{align} and so $f(x) = f^{-1}(y)$ only at $x = 0$ and $x = 1$, but at these points $f(x) \neq x$.

• Consider $f$ is continuous Feb 23 '17 at 17:24
• I will have a think @S.H.W, I imagine the answer might be that such a function cannot exist: if they intersect at two points then they are equal on an interval (that is my hunch anyway, YMMV). Feb 23 '17 at 21:17
• I think this statement is true but I can't prove it. Feb 23 '17 at 21:32
• Actually, I don't think that is quite true: consider $f(x) = x^2$ for $x \geqslant 0$ and $f(x) = -2x$ for $x < 0$, then $f^{-1}(x) = \sqrt{x}$ for $x \geqslant 0$ and $f^{-1}(x) = -\tfrac{x}{2}$ for $x < 0$ and so they only intersect at $x = 0$ and $1$. Feb 23 '17 at 21:59
• if $f(x) = x^2$ then $f^{-1}(x) = \sqrt{x} , x \ge 0$ and the intersection points are $(0,0)$ and $(1,1)$ which they are in $y=x$ line Feb 23 '17 at 22:03

For $f$ and $f^{-1}$ to intersect there must be a point(s) $(x,f(x)) = (x,f^{-1}(x))$. Assume $x \neq f(x)$. Because $f$ and $f^{-1}$ are symmetric around y=x then there must be a corresponding point(s) $(f(x),x) = (f^{-1}(x),x)$.

The line defined by the points $(x_0,f(x_0)$ and $(f(x_),x_0)$ is $$(y-f(x_0) = \frac{f(x_0)-x_0}{x_0-f(x_0)}(x-x_0)$$ and simplified becomes $$y=-x+x_0+f(x_0)$$ This line, with a slope of -1, is not parallel with $y=x$, therefore must cross $y=x$. Because the two points are symmetric around $y=x$, they must lie on different sides of $y=x$. By the intermediate value theorem both $f$ and $f^{-1}$ must cross $y=x$.

Hence there is at least one point of $f$ on $y=x$


# A particular vanishing integral

While dealing with a definite integral on AoPS I discovered (I have to admit by pure chance) the following relation

$$\int_0^1\log\left(\frac{(x+1)(x+2)}{x+3}\right)\frac{\mathrm dx}{1+x}~=~0\tag1$$

The proof is quite easy, but feels kind of contrived. Indeed, just apply a self-similar substitution - $$x\mapsto\frac{1-x}{1+x}$$ - to the auxiliary integral $$\mathcal I$$ given by

$$\mathcal I~=~\int_0^1\log\left(\frac{x^2+2x+3}{(x+1)(x+2)}\right)\frac{\mathrm dx}{1+x}$$

And the result follows. However, to consider precisely this integral seems highly unnatural to me (in fact, as I mentioned earlier, this integral was just a by-product while evaluating something quite different and I discovered $$(1)$$ when experimenting with various substitutions).

The crucial point to notice concerning $$\mathcal I$$ is the invariance of the polynomial $$f(x)=x^2+2x+3$$ regarding the self-similar substitution which allows us to deduce $$(1)$$. Additionally for myself I am quite surprised by the special structure of $$(1)$$ since we have factors of the form $$(x+1)$$, $$(x+2)$$ and $$(x+3)$$ combined which calls for a generalization (although I found none yet).

It there a more elementary approach, not relying on such an "accident" like examining the integral $$\mathcal I$$ for proving $$(1)$$? Additionally, can this particular pattern be further generalized? Answers to both questions (also separately) are highly appreciated!

• The $\displaystyle\log$ argument numerator -in the $\displaystyle\mathcal{I}$ definition- must be $\displaystyle x^{2} + 3x + 2$ instead of $\displaystyle x^{2} + 2x + 3$. – Felix Marin Aug 27 '20 at 3:27
• @FelixMarin No. By my calculations $\left(\frac{1-x}{1+x}\right)^2+2\left(\frac{1-x}{1+x}\right)+3=2\frac{x^2+2x+3}{(1+x)^2}$ which is what the approach is all about. But $\left(\frac{1-x}{1+x}\right)^2+3\left(\frac{1-x}{1+x}\right)+2=2\frac{x+3}{(x+1)^2}$ which yields an interesting other identity but not the one the post is concerned with. – mrtaurho Aug 27 '20 at 3:42

That's quite an impressive method to show that the integral vanishes.

For the first part I'll show using a different approach that your integral vanishes. $$\mathcal J=\int_0^1 \ln\left(\frac{x+3}{(x+2)(x+1)}\right)\frac{\mathrm dx}{x+1}\overset{x+1=t}= \color{blue}{\int_1^2\ln\left(\frac{t+2}{t+1}\right)\frac{\mathrm dt}{t}}-\color{red}{\int_1^2 \frac{\ln t}{t}\mathrm dt}$$ Let's denote the blue integral as $$\mathcal J_1$$ then using the substitution $$\frac{2}{t}\to t$$ we get: $$\mathcal J_1=\int_1^2 \ln\left(\frac{t+2}{t+1}\right)\frac{\mathrm dt}{t}=\int_1^2 \ln\left(\frac{2(t+1)}{t+2}\right)\frac{\mathrm dt}{t}$$ Adding both integrals from above gives us: $$\require{cancel} 2\mathcal J_1=\cancel{\int_1^2 \ln\left(\frac{t+2}{t+1}\right)\frac{\mathrm dt}{t}}+\int_1^2 \frac{\ln 2}{t}\mathrm dt+\cancel{\int_1^2 \ln\left(\frac{t+1}{t+2}\right)\frac{\mathrm dt}{t}}=\ln^2 2$$ $$\Rightarrow \mathcal J_1=\color{blue}{\frac{\ln^2 2}{2}}\Rightarrow \mathcal J=\color{blue}{\frac{\ln^2 2}{2}}-\color{red}{\frac{\ln^2 2}{2}}=0$$

As for the second part, a small generalization outcomes by experimenting with the blue integral.

In particular, by the same approach we have: $$\int_1^a \ln\left(\frac{x+a}{x+1}\right)\frac{\mathrm dx}{x}=\int_1^a \frac{\ln x}{x}\mathrm dx$$ Which gives us a small generalization: $${\int_0^{a-1}\ln\left(\frac{x+a+1}{(x+1)(x+2)}\right)\frac{\mathrm dx}{x+1}=0}$$ Similarly, (with the substitution $$\frac{ab}{x}\to x$$) we get that: $$\int_a^b \ln\left(\frac{x+b}{x+a}\right)\frac{dx}{x}=\frac12 \ln^2 \left(\frac{b}{a}\right)=\int_a^b \ln\left(\frac{x}{a}\right)\frac{dx}{x}$$ And the following follows: $$\int_{a-1}^{b-1} \ln\left(\frac{a(x+b+1)}{(x+1)(x+a+1)}\right)\frac{dx}{x+1}=0$$ One might be interested in the following similar generalization too: $$\int_1^{t}\ln\left(\frac{x^4+sx^2+t^2}{x^3+sx^2+t^2x}\right)\frac{dx}{x}=0,\quad s\in R, t>1$$

• This is what I was looking for; I guess^^ (+1) anyway and I'm looking forward to see a generalization (if possible). I guess you know where I got this integral from :D – mrtaurho Aug 12 '19 at 15:17
• @mrtaurho I got there a small generalization for now. However since $\int_a^b \ln\left(\frac{x+b}{x+a}\right)\frac{dx}{x}=\frac12 \ln^2 \left(\frac{b}{a}\right)$ it might be possible to obtain a better one. (I'll try later to work with it). – Zacky Aug 12 '19 at 15:38
• It's hard to write out your name now :P But yes, this seems promising, I'm curious! As I noted above it was a rather strange by-product to discover this equality; and it was tedious to ran into it three times while hoping for something more helpful for solving the original task^^' – mrtaurho Aug 12 '19 at 15:44
• @mrtaurho just in case you missed it in winter I'll mention that $\mathcal I$ (before the self-similar sub was applied) originates from this generalization: math.stackexchange.com/a/3049039/515527. Aka: $$\int_1^{\sqrt{t}}\ln\left(\frac{x^4+sx^2+t}{x(x^2+sx+t)}\right)\frac{dx}{x}=0$$ – Zacky Aug 12 '19 at 16:06
• As I've upvoted both, the question you linked and your answer, I guess I've seen it at some point. But I'll take a look at it again :) – mrtaurho Aug 12 '19 at 16:23

I have used the substitution $$(x+1)(y+1)=2$$ before to good effect because $$\int_0^1f(x)\,\frac{\mathrm{d}x}{x+1}=\int_0^1f\!\left(\tfrac{1-y}{1+y}\right)\frac{\mathrm{d}y}{y+1}\tag1$$ If $$f(x)=\log\left(\frac{x+3}{(x+2)(x+1)}\right)$$, then $$f\!\left(\frac{1-y}{1+y}\right)=\log\left(\frac{(y+2)(y+1)}{y+3}\right)$$. Therefore $$\int_0^1\log\left(\frac{x+3}{(x+2)(x+1)}\right)\frac{\mathrm{d}x}{x+1}=\int_0^1\log\left(\frac{(y+2)(y+1)}{y+3}\right)\frac{\mathrm{d}y}{y+1}\tag2$$ and since the two sides of $$(2)$$ are negatives, they are both $$0$$.
We can generalize $$(1)$$ by letting $$(x+a)(y+a)=a(1+a)$$, then we get $$\int_0^1f(x)\frac{\mathrm{d}x}{x+a}=\int_0^1f\!\left(\tfrac{a(1-y)}{a+y}\right)\frac{\mathrm{d}y}{y+a}\tag3$$