Problem: Expected value of a continuous random variable

Problem

\(X\) is a continuous random variable with probability density function \(f\left( x \right)=3x^2\) for \(0≤x≤1\) ( \(f\left( x \right)=0\) otherwise). Find \(E(X)\) .

Solution

\[E\left( X \right)=\int_{0}^{1}{ xf\left( x \right) }dx=\int_{0}^{1}{ 3x^3 }dx={\left[ \frac{3}{4}x^4 \right]}_0^1=3/4\]