Chapter 4: Generating Functions - Auckland
The probability generating function gets its name because the power series can be expanded and differentiated to reveal the individual probabilities. Thus, given only the PGFGX(s) = E(sX), we can recover all probabilitiesP(X = x). For shorthand, write px = P(X = x). Then GX(s) = E(sX) = X∞ x=0 pxs x = p 0+ p1s + p2s 2+p 3s 3+ p 4s 4+ ...
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