Question

Suppose X~Binomial(n,p).Find the pgf of Y=X+1 in terms of GP(x)

Answer 1

The probability generating function (PGF) of X is G_X(s) = E(s^X), for all s ∈ R for which the sum converges.

for Binomial Distribution:

$G_X(s)=\sum s^x\begin{pmatrix} n \\ x \end{pmatrix}p^xq^{n-x}=\sum \begin{pmatrix} n \\ x \end{pmatrix}(ps)^xq^{n-x}=(ps+q)^n$

$G_Y(s)=\sum \begin{pmatrix} n \\ y \end{pmatrix}(ps)^yq^{n-y}=\sum \begin{pmatrix} n \\ x+1 \end{pmatrix}(ps)^{x+1}q^{n-(x+1)}=(ps+q)^n=G_X(s)$