Expected value (continuous random variable)

$E[X]=\int x\,f(x)\,dx$ — the long-run average value $X$ takes, weighted by its density.

The expected value of a continuous random variable $X$ with density $f(x)$ is $E[X]=\int x\,f(x)\,dx$ — a weighted average of every possible value $x$ , weighted by how likely values near $x$ are (given by $f(x)$ ). It's the continuous analogue of a discrete weighted average, generalizing "sum of value times probability" to "integral of value times density."

Examples

For $f(x)=\frac{2x}{b^2}$ on $[0,b]$ , $E[X]=\int_0^b x\cdot\frac{2x}{b^2}\,dx=\dfrac{2b}{3}$ .