Expected Value

Study guides: Exam P-1, Exam MAS-I, Exam MAS-II

Expected Value ( $E[X]$ , also called the mean $\mu$ ) is the probability-weighted average of all values a Random Variable $X$ can take, representing its long-run average outcome.

$E[X] = \begin{cases} \displaystyle\sum_{k} k\, f(k) & \text{(discrete)} \\[6pt] \displaystyle\int_{-\infty}^{\infty} x\, f(x)\, dx & \text{(continuous)} \end{cases}$

The sum or integral must converge absolutely ( $\sum_k |k|\,f(k)$ or $\int |x|\,f(x)\,dx$ finite); otherwise $X$ has no expected value — a Pareto with shape $\alpha \le 1$ has none (see Moment).
It is linear: $E[aX + b] = aE[X] + b$
The $n$ -th moment of $X$ is $E[X^n]$ , and the $n$ -th central moment is $E[(X-\mu)^n]$

Expected Payout on a Simple Policy A discrete random variable $X$ (claim size) has PMF: $P(X=0)=0.5$ , $P(X=100)=0.3$ , $P(X=500)=0.2$ . Find $E[X]$ .

Answer $E[X] = 0(0.5) + 100(0.3) + 500(0.2) = 0 + 30 + 100 = 130$ On average, the insurer expects to pay $130 per claim.