By James E. Gentle

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15) is over an abstract domain Ω. We can also write the expectation over the real range of the random variable and an equivalent measure on that range. 16) IR d or in the more precise form, E(X) = x dF (x). IRd If the PDF exists and is f, we also have E(X) = xf(x) dx. IRd An important and useful fact about expected values is given in the next theorem whose proof is an exercise. 7 Let X be a random variable in IRd such that E( X E(X). Then aµ = arg min E( X − a 2 ). 17) a∈IR d In statistical applications this result states that E(X) is the best prediction of X given a quadratic loss function in the absence of additional information about X.

Random variables X1 , . . , Xn such that d X = X1 + · · · + Xn . Notice that (n + 1)-divisibility does not imply n-divisibility. We can see this in the example of a random variable X with a binomial (3, π) distribution, which is clearly 3-divisible (as the sum of three Bernoullis). d. X1 and X2 such that A Companion for Mathematical Statistics c 2010 James E. 1 Some Important Probability Facts 43 X = X1 + X2 , because if there were, each Xi would take values in [0, 3/2] with Pr(Xi = 0) > 0 and Pr(Xi = 0) > 0, but in that case Pr(X = 3/2) > 0.

Then aµ = arg min E( X − a 2 ). 17) a∈IR d In statistical applications this result states that E(X) is the best prediction of X given a quadratic loss function in the absence of additional information about X. We define the expected value of a Borel function of a random variable in the same way as above for a random variable. 18 (expected value of a Borel function) If g is a Borel function of the random variable X, then the expected value of g(X) is A Companion for Mathematical Statistics c 2010 James E.