Expectation for Continuous RVs

Probability · Axiom Academy

LESSON Expectation for Continuous RVs Computing the average value using integration The expected value E[X] of a continuous random variable X with PDF f(x) is defined as: This integral weights each value x by its density f(x) and sums across all possible values. Think of it as the "center of mass" or "balance point" of the probability distribution. To find the expected value of a function g(X), we don't need to find the PDF of g(X) first! Instead, we use the Law of the Unconscious Statistician (LOTUS): This formula allows us to compute E[g(X)] directly using the original PDF of X. Linearity: E[aX + b] = aE[X] + b for constants a and b. Sum rule: E[X + Y] = E[X] + E[Y], even if X and Y are dependent. Product rule (independence): If X and Y are independent, then E[XY] = E[X]·E[Y].

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