factorial moment generating function

Question

I'm trying to get the factorial moment-generating function of a binomial random variable. I know that

$F_X(t) = E[t^x] = \Sigma_xt^xp(x)$

so I get $\Sigma_xt^x{n \choose x}\theta^x(1-\theta)^1-x$

where $\theta$ being the probability of a success.

I can't expand this equality to get the actual equation I need (my calculation isn't very creative). Any help here?

Answer 1

The sum is $(\sum \binom {n} {x} s^{x}(1-\theta)^{n}$ where $s =\frac {t\theta} {1-\theta}$. Hence it is $(1-\theta)^{n} (1+s)^{n}=(1-\theta)^{n} (1+\frac {t\theta} {1-\theta})^{n}=(t\theta +(1-\theta))^{n}$.