Answer to Question #277889 in Statistics and Probability for Johnnie

Question

Let X1,X2,...,Xn be independent and identifically distributed random variables each having pdf

f(x)=q^xP .x=0,1,2....

P+q=1

Let Sn=X1+X2+...+Xn.

Obtain the pgf of Sn and find the mean and variance of Sn

Answer 1

X₁,X₂,...,X_n be IID.

$f(x)=q^xp;,x=0,1,2,... \\X\sim G(p) \\E(X)=\dfrac qp, Var(X)=\dfrac q{p^2} \\ M_X(t)=\dfrac p{1-qe^t}, G_X(t)=\dfrac p{1-qt} \\S_n=X_1+X_2+...+X_n \\ G_{S_n}(Z)=G_{X_1}(Z)G_{X_2}(Z)...G_{X_n}(Z)=\Pi_{i=1}^nG_{X_i}(Z) \\G_X(Z)=E(Z^x)=\dfrac p{1-qz}\ \ \ \ \ \ [G_X(.)\rightarrow pdf\ of\ X]$

So, $G_{S_n}(Z)=(\dfrac p{1-qz})^n$

$E(S_n)=E(X_1+X_2+...+X_n) \\=\Sigma_{i=1}^nE(X_i) \\=\Sigma_{i=1}^n \dfrac qp \\=\dfrac {nq}p$

$Var(S_n)=Var(X_1+X_2+...+X_n) \\=\Sigma_{i=1}^nVar(X_i)$ [Since X_i's are IIDs]

$=n\times \dfrac q{p^2} \\=\dfrac {nq}{p^2}$