EXPECTATIONS

Expectations. (See also Hays, Appendix B; Harnett, ch. 3). A. The expected value of a random variable is the arithmetic mean of that variable, i.e. E(X) = μ. As Hays notes, the idea of the expectation of a random variable began with probability theory in games of chance. Gamblers wanted to know their expected long-run winnings (or losings) if they played a game repeatedly. This term has been retained in mathematical statistics to mean the long-run average for any random variable over an indefinite number of trials or samplings. B. Discrete case: The expected value of a discrete random variable, X, is found by multiplying each X-value by its probability and then summing over all values of the random variable. That is, if X is discrete,