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    IBHardProbability10–12

    When Expectation Passes Through a Function and When It Does Not

    Quando o Valor Esperado Atravessa uma Função e Quando Não Atravessa

    A discrete random variable X takes the values 1, 2 and 3 with probabilities \frac{1}{2}, \frac{1}{3} and \frac{1}{6}. Find E(X), E(3X - 1) and E\left(X^{2}\right), and hence \operatorname{Var}(X). Compare E\left(X^{2}\right) with \left(E(X)\right)^{2} and explain what the comparison shows.

    The expectation of a function of X is found by applying the function to each value and weighting by the same probabilities. Whether that agrees with applying the function to E(X) depends on the function, and the two functions here behave differently.

    Solution

    Step 1 of 7

    1. 1.E(X) = 1 \times \frac{1}{2} + 2 \times \frac{1}{3} + 3 \times \frac{1}{6} = \frac{1}{2} + \frac{2}{3} + \frac{1}{2} = \frac{5}{3}