Answer to Question #136612 in Statistics and Probability for Elaine Mabote

Question

Let X~Bin(n,p).Find E(e^tx) where t is a constant

Answer 1

Using independence, Ee^tΣXi=E$\Pi$ e^tXi=$\Pi$Ee^tXi=(pe^t+(1-p))ⁿ,

where the Xi are independent Bernoulli random variables. Equivalently

$\sum^n_{k=0}e^{tk}(^n_k)p^k(1-p)^{n-k}=\sum^n_{k=0}(^n_k)(pe^{t})^k(1-p)^{n-k}=(pe^t+(1-p)^n)$ by the Binomial formula.