By James E. Gentle

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Xk on (Ω, F , P ) are said to be independent iff for any sets B1 , . . , Bk in the σ-field of the image space, k P (X1 ∈ B1 , . . , Xk ∈ Bk ) = i=1 P (Xi ∈ Bi ). 1 on page 4). If X1 and X2 are random variables with joint PDF fX1 ,X2 a joint PDF and marginal PDFs fX1 and fX2 , then X1 and X2 are independent iff fX1 ,X2 (x1 , x2 ) = fX1 (x1 )fX2 (x2 ). 16 (exchangeability of random variables) The random variables X1 , . . , Xk on (Ω, F , P ) are said to be exchangeable iff the joint distribution of X1 , .

For a discrete distribution, this expression can be interpreted as a sum of the values at the mass points times their associated probabilities. V(·) The variance of X is denoted by V(·). , we see that the variance is nonnegative. s. Further, unless X = E(X), the variance is positive. The square root of the variance is called the standard deviation. If it exists, E(|X|) is called the first absolute moment of X; and generally, if it exists, E(|X|r ) is called the r th absolute moment. 10 If the r th absolute moment of a scalar random variable exists (that is, it is finite), then the absolute moments of order 1, 2, .

The frequency-generating function or probability-generating function is the polynomial P (t) = ∞ pi+1 ti . 64) The probability-generating function for the binomial distribution with parameters π and n, for example, is P (t) = (πt + (1 − π))n . A discrete distribution with support x1 , x2 , . . is equivalent to a discrete distribution with support 0, 1, . .. For a discrete distribution with support 0, 1, . . another kind of moment is sometimes useful. It is the factorial moment, related to the r th factorial of the real number y: y[r] = y(y − 1) · · · (y − (r − 1)).

### A Companion for Mathematical Statistics by James E. Gentle

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