Given X∼Binomial(n,p)X \sim \text{Binomial}(n, p)X∼Binomial(n,p), find the MGF MX(t)=E[etX]M_X(t) = E[e^{tX}]MX(t)=E[etX].
pent+1−pp e^{nt} + 1 - ppent+1−p
enp(et−1)e^{np(e^t-1)}enp(et−1)
(pet+1−p)n(p e^t + 1 - p)^n(pet+1−p)n
(q+pet)n(q + p e^t)^n(q+pet)n where q=1−pq=1-pq=1−p