By James E. Gentle
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Additional info for A Companion for Mathematical Statistics
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.