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Expected Value

High School

Definition

Expected value (mean) is the probability-weighted average of all possible values. It represents the long-run average.

E[X] = ∑ᵢ xᵢ · P(X = xᵢ)

Expected value (discrete)

E[X] = ∈t_-∈fty^∈fty x · f(x) dx

Expected value (continuous)

E[aX + b] = aE[X] + b

Linearity of expectation

Example 1

Find the expected value when rolling a die.

Example 2

A lottery wins $1000 with P=1/1000, else $0. Find E[X].

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