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Concept

Expected Value

Expected value is the long-run average outcome of a random variable, not necessarily the most likely outcome.

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Formula

For a discrete random variable, expected value is a probability-weighted average. For a continuous random variable, the sum becomes an integral against the density.

Discrete expected value
E[X]=∑xx P(X=x)E[X] = \sum_x x\,P(X=x)
Continuous expected value
E[X]=∫−∞∞xfX(x) dxE[X] = \int_{-\infty}^{\infty} x f_X(x)\,dx

Worked Example

Suppose a claim is 0 with probability 0.70, 100 with probability 0.20, and 500 with probability 0.10. The expected claim is 0(0.70)+100(0.20)+500(0.10)=70.

Exam P Trap

The most common mistake is replacing E[g(X)] with g(E[X]). That only works in special linear cases.

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