Expectation

Definition

Continuous random variable

Let

• $X$ : Continuous random variable
• $f(x)$ : pdf of $X$
• ${\int }_{-\infty }^{\infty }|x|f(x)dx<\infty$

If $E(X)={\int }_{-\infty }^{\infty }xf(x)dx$

Then we say $E(X)$ is the expectation of $X$

Discrete random variable

Let

• $X$ : Discrete random variable
• $p(x)$ : pmf of $X$
• ${\sum }_x|x|p(x)<\infty$

If $E(X)={\sum }_{x}xp(x)$

Then we say $E(X)$ is the expectation of $X$

Remark

Sometimes $E(X)$ is called the mathematical expectation, expected value, or mean of $X$. When “mean” is used, we often denote $E(X)=\mu$.

Related theorems

• Expectation of a function
• Linearity of expectation
• Existence of lower order moments