Probability generating functions

• If a discrete random variable X has probability mass function P(X = x) then the probability generating function of X is Gx(t) = $\sum$ P(X = x) t ^x.
• The PGF of a DRV X ~ B (n, p) is Gx(t) = (1 - p + p t) ^ n. (in FB)
• The PGF of a DRV X ~ NB (r, p) is Gx(t) = (p t/1 - (1 - p) t) ^ r. (in FB)
• The PGF of a DRV X ~ Po($\lambda$ ) is Gx(t) = e ^ $\lambda$ (t - 1). (in FB)
• The PGF of a DRV X ~ Geo(p) is Gx(t) = p t/1 - (1 - p) t. (in FB)
• The mean of a PGF is the differential: E(X) = G' x(1). (in FB)
• The variance of a PGF is Var(X) G'' x(1) + G' x(1) - (G' x (1)) ^2. (in FB)
• G ^n x(0) = P(X = n). (NOT in FB)
• If X and Y have PGFs Gx(t) and Gy(t) then the PGF of Z = X+ Y is Gz(t) = Gx(t) x Gy(t). (in FB)
• If X has a PGF of Gx(t), the PGF of Y = aX + b is Gy(t) = t^b Gx(t^a). (NOT in FB)