## Definition The ==Expected Value== (or mean) of a random variable $X$ is the probability-weighted average of all possible values. For a discrete random variable: $ E[X] = \sum_{x} x \cdot P(X = x) $ For a continuous random variable: $ E[X] = \int_{-\infty}^{\infty} x \cdot f(x)\, dx $ > [!example]- An insurance policy pays $\$0$ with probability 0.8, $\$500$ with probability 0.15, and $\$5{,}000$ with probability 0.05. What is the expected payout? > $ E[X] = 0(0.8) + 500(0.15) + 5000(0.05) = 0 + 75 + 250 = \$325 $