“probability generating function”的概念、定义、翻译、参考文献-科学参考

probability generating function

Mathematics

单词	probability generating function
释义	probability generating function Mathematics If X is a random variable where X takes only non-negative integer values, then G_X(s) = E(s^X) is the probability generating function (pgf). Then $G_{X} (s) = \sum_{k = 0}^{\infty} Pr (X = k) s^{k}$ The mean and variance of X can also be computed using the pgf from $E (X) = G_{X}^{'} (1) and Var (X) = G_{X}^{″} (1) + G_{X}^{'} (1) - {(G_{X}^{'} (1))}^{2} .$ The pgf of the sum X + Y of two independent random variables is given by $G_{X + Y} (s) = G_{X} (s) G_{Y} (s),$ and for random sampling, where Z = X₁ + … + X_N, with N itself a random variable and the X_i are iid, then $G_{Z} (s) = G_{N} (G_{X} (s)) .$
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