Expected value

Definition

Let $X$ be a real random variable on a probability space $(\Omega, \mathcal{F}, P)$. If $X \ge 0$, the expected value of $X$ is the Lebesgue integral $$E[X] = \int_\Omega X \, dP \in [0, \infty].$$ In general, write $X = X^+ - X^-$ with $X^+ = \max(X, 0)$ and $X^- = \max(-X, 0)$. If $E[|X|] < \infty$, then $X$ is integrable and $E[X] = E[X^+] - E[X^-]$.

Remarks

For a discrete random variable with values $x_1, x_2, \dots$, $E[X] = \sum_i x_i \, P(X = x_i)$ when the series converges absolutely. For a random variable with density $f$, $E[X] = \int_{-\infty}^{\infty} x f(x) \, dx$ when $\int |x| f(x) \, dx < \infty$. Other notations are $\mathbb{E}[X]$, $\mathbb{E}X$ and $\mu_X$. The expected value depends only on the law of $X$. If $E[X^+] = \infty$ and $E[X^-] < \infty$, one sets $E[X] = +\infty$. If both are infinite, $E[X]$ is not defined.

Explanations

L1 human

The expected value is the long-run average of many independent repetitions of the experiment. For a fair die, it is $(1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5$. This value never appears on the die, but it is the center of mass of the distribution.