Question

Consider the geometric random variable X with probability mass function P(X =x)=(1 p)x 1p, x=1,2,3,....

For t <- l o g ( 1 - p ) , c o m p u t e E [ e t X ] .

Answer 1

E(e^tX) = Sigma_x = 1^infinity e^tx P(X = x) = Sigma_x = 1^infinity e^tx(1 - p)^x - 1 p = p/1 - p Sigma_x = 1^infinity e^tx (1 - p)^x = p/1 - p Sigma_x = 1^infinity [(1 - p) e^t]^x = p/1 - p[(1 - p) e^t + ((1 - p)e^t)^2 + ((1 - p) e^t)^3 + ...] = p/1 - p times (1 - p) e^t[1 + (1 - p) e^t + ((1 - p) e^t^2 + ...] = pe^t[1 - (1 - p)e^t]^-1 (Because 1 + x + x^2 + ... = (1 - x)^-1) Rightarrow E(e^tX) = pe^t/1 - (1 - p) e^t