5.2 Expected Values, Covariance, and Correlation

content

• 5.2.1 Covariance
• 5.2.2 Correlation
• 5.2.3 The Bivariate Normal Distribution
• EXERCISES Section 5.2 (22-36)

introduction

Any function $h\left(X\right)$ of a single rv $X$ is itself a random variable. However, we saw that to compute $E\left[h\left(X\right)\right]$, it is not necessary to obtain the probability distribution of $h\left(X\right)$. Instead, $E\left[h\left(X\right)\right]$ is computed as a weighted average of $h\left(x\right)$ values, where the weight function is the pmf $p\left(x\right)$ or $\mathrm{pdf}f\left(x\right)$ of $X$. A similar result holds for a function $h\left(X,Y\right)$ of two jointly distributed random variables.

Proposition (expected value of h(X,Y))

Let $X$ and $Y$ be jointly distributed rv’s with $\mathrm{pmf}p\left(x,y\right)$ or $\mathrm{pdf}f\left(x,y\right)$ according to whether the variables are discrete or continuous. Then the expected value of a function $h\left(X,Y\right)$, denoted by $E\left[h\left(X,Y\right)\right]$ or ${\mu }_{h\left(X,Y\right)}$, is given by

$$\{\begin{array}{ll}\sum _x\sum _{y}h\left(x,y\right)\cdot p\left(x,y\right)& \text{ if }X\text{ and }Y\text{ are discrete }\\ {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }h\left(x,y\right)\cdot f\left(x,y\right)dxdy& \text{ if }X\text{ and }Y\text{ are continuous }\end{array}$$Link to original

EXAMPLE 5.13

EXAMPLE 5.14

The method of computing the expected value of a function $h\left(X_1,\dots ,X_n\right)$ of $n$ random variables is similar to that for two random variables. If the $X_i$ ’s are discrete, $E\left[h\left(X_1,\dots ,X_n\right)\right]$ is an $n$ -dimensional sum; if the $X_i$ ’s are continuous, it is an $n$ - dimensional integral.