By James E. Gentle

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**Extra resources for A Companion for Mathematical Statistics**

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If X is an integrable random variable over A, then IA (X) is an integrable random variable, and Pr(A) = E(IA (X)). 23) When it is clear from the context, we may omit the X, and merely write E(IA ). Expected Value over a Measurable Set The expected value of an integrable random variable over a measurable set A ⊂ IRd is E(XIA (X)) = X dP. 24) A We often denote this as E(XIA ). Expected Value of General Measurable Functions A real-valued measurable function g of a random variable X is itself a random variable, possibly with a different probability measure.

In the change of variable method, we think of h as a mapping of the range X of the random variable X to the range Y of the random variable Y , and the method works by expressing the probability content of small regions in Y in terms of the probability content of the pre-image of those regions in X . For a given function h, we often must decompose X into disjoint sets over each of which h is one-to-one. 10 distribution of the square of a standard normal random variable Suppose X ∼ N(0, 1), and let Y = h(X) = X 2 .

17 (expected value of a random variable) Given a probability space (Ω, F , P ) and a d-variate random variable X defined on F , we define the expected value of X with respect to P , which we denote by E(X) or for clarity by EP (X), as E(X) = X dP. Ω A Companion for Mathematical Statistics c 2010 James E. 15) 16 1 Probability and Distribution Theory For the random variable X, E(X), if it exists, is called the first moment of X. Although by definition, Pr(−∞ < X < ∞) = 1 it could happen that X dP does not exist (or bending the notation slightly, Ω X dP = ±∞).

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