Expected value of a discrete random variable

Summary

Definition

• If $X$ is a discrete random variable with range ${x_1,x_2,…,x_n}$ and probability mass function $f(x)$ then the expected value of $X$ is defined as

$$E\left( X \right)=\sum_{i=1}^{n}{ x_if(x_i) }$$

Example

• $X$ is a random variable which can take values 1,2 or 3 and has a probability mass function given by
• $f(1)=0.4$ ,
• $f(2)=0.3$ and
• $f(3)=0.3$ .
• The expected value of $X$ is

$$E(X)=1×0.4+2×0.3+3×0.3=1.9$$