CH39 - Variance of Sums, Covariance, Correlations

Essential reasoning for this chapter is that for expected value calculations:

$$E(X_1+\cdots +X_n)=E(X_1)+\cdots +E(X_n)$$

Meaning Random Variables can be independent or dependent.

However, for Variance calculations:

$$VAR(X_1+\cdots +X_n)=VAR(X_1)+\cdots +VAR(X_n)$$

The Random Variables HAVE to be independent.

This chapter will explore what we can do if we have dependent random variables.

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IF $x,y$ are dependent, then

$$VAR(X+Y)=VAR(X)+VAR(Y)+2Cov(X,Y)$$

Essentially, the covariance term is a measure of how much $X$ and $Y$ vary together.

Definition

Covariance is a measure of the joint variability of two random variables. It is defined as:

$$Cov(X,Y)=E\left[(X-E(X))(Y-E(Y))\right]=E(XY)-E(X)E(Y)$$

This definition can be used for the following formula:

For any two random variables $X$ and $Y$, the covariance is given by:

$$Cov(X,X)=VAR(X)$$

For the variance of the sum of two random variables, we have:

$$VAR(X_1+\cdots +X_n)=\sum _{i=1}^{n}var(X_i)+\sum _{i=1}^n\sum _{j+i}Cov(X_i,X_j)$$

The covariance of two random variables $X$ and $Y$ is a measure of how much they change together. It is defined as:

$$Cov(X,Y)=Cov(Y,X)$$

And if $X$ and $Y$ are independent, then:

$$If:X,Yindependent,then:Cov(X,Y)=0$$

We can also measure the correlation, p, as the ratio of the covariance to the product of the standard deviations of $X$ and $Y$:

$$P(X,Y)=\frac{Cov(X,Y)}{\sqrt{VAR(X)\cdot VAR(Y)}}$$

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Exercise 39.13

Roll two dice. Let $X$ denote the maximum value that appears and let $Y$ denote the minimum value that appears.

$\tau$

Find $Cov(X,Y)$:

$$E(X)\theta (Y)E(XY)E(x^2)$$

$x$	$P_X(x)$	$y$	$P_Y(y)$	$xy$	$P(XY)(x,y)$
1	$\frac{1}{36}$	1	$\frac{11}{36}$
2	$\frac{3}{36}$	2	$\frac{9}{36}$
3	$\frac{5}{36}$	3	$\frac{7}{36}$
4	$\frac{7}{36}$	4	$\frac{5}{36}$
5	$\frac{9}{36}$	5	$\frac{3}{36}$
6	$\frac{11}{36}$	6	$\frac{1}{36}$