Expected value of a continuous random variable

Summary

Definition

• If $X$ is a continuous random variable with range $\left[ a,b \right]$ and probability density function $f(x)$ then the expected value of $X$ is defined as

$$E\left( X \right)=\int_{a}^{b}{ xf\left( x \right)dx }$$

• Note: $a$ may be $-∞$ and/or $b$ may be $∞$ .

Example

• $X~U[0,1]$ . Then $E(X)=\int_{0}^{1}{ ‍ }xf(x)dx=\int_{0}^{1}{ ‍ }xdx=0.5$ .

Symmetric pdf

• If the probability density function of $X$ is symmetric around some constant $b$ then $E(X)=b$ .

Example

• $Z$ is a s t andard normal random variable . The pdf is symmetric around 0 and $E(Z)=0$ .