Expected Values of Continuous Random Variables

For a discrete random variable, the expected value is a weighted sum of outcomes weighted by their probabilities. For a continuous random variable, we replace the sum with an integral and the probability mass function with the probability density function.

The expected value of a continuous random variable $X$ with probability density function $f(x)$ is defined as

$$E[X] = \int_{-\infty}^{\infty} x\, f(x)\, dx.$$

If $f(x) = 0$ outside an interval $[a,b]$, the formula simplifies to

$$E[X] = \int_a^b x\, f(x)\, dx.$$

For example, suppose $X$ has density $f(x) = 1$ for $0 \le x \le 1$ (and $0$ otherwise). Then

$$E[X] = \int_0^1 x \cdot 1\, dx = \left[\dfrac{x^2}{2}\right]_0^1 = \dfrac{1}{2}.$$

This is exactly what we expect: the average value of a uniformly distributed point on $[0,1]$ is the midpoint $\tfrac{1}{2}$.