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 \(E(Z)=0\) .