By James E. Gentle
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Additional info for A Companion for Mathematical Statistics
1 Some Important Probability Facts E(g(X)) = g(x) dF (x). s. 19) E(g(X)) ≥ 0. s. ⇒ Proof. Each property is an immediate consequence of the definition. Expected Value and Probability There are many interesting relationships between expected values and probabilities. , then ∞ E(X) = Pr(X > t)dt. 21) 0 Proof. 9 leads in general to the useful property for any given random variable X, if E(X) exists: ∞ E(X) = 0 0 (1 − F (t))dt − F (t)dt. 22) −∞ Another useful fact in applications involving the Bernoulli distribution with parameter π is the relationship E(X) = Pr(X = 1) = π.
To emphasize the meaning more precisely, we could write the integral in the definition as E(X) = X(ω) dP (ω). 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).
The self-information of X = x is − log2 (pX (x)). The logarithm to the base 2 comes from the basic representation of information in base 2, but we can equivalently use any base, and it is common to use the natural log in the definition of self-information. The logarithm of the PDF is an important function in studying random variables. It is used to define logconcave families (see page 101) and its derivative with respect to a parameter of a distributional family appears in an important inequality in statistical theory (see pages 267 and 648).
A Companion for Mathematical Statistics by James E. Gentle