# Moment generating function (find the probability)

The moment generating function of a random variable $$X$$ is given by:

$$M(t) = (1/3^{2k})(7+2e^t)^k, \quad \forall t$$

a) Determine $$P(X = 3)$$

b) Derive the $$r^{th}$$ factorial moment of $$X$$

I assume you must deduce what distribution this is from the mgf, I just have no idea which one it is.

Let $$X_i\stackrel{\mathrm{i.i.d.}}\sim\mathsf{Ber}(p)$$ random variables for $$i=1,2,\ldots$$, then the moment generating function of $$X_1$$ is $$\mathbb E[e^{tX_1}] = 1-p + pe^t.$$ Set $$S_k = \sum_{i=1}^k X_i$$, $$k=1,2,\ldots$$. Then the moment generating function of $$S_k$$ is given by \begin{align} \mathbb E[e^{tS_k}] &= \mathbb E\left[e^{t\sum_{i=1}^k X_i} \right]\\ &= \prod_{i=1}^k \mathbb E[e^{tX_i}]\\ &= \prod_{i=1}^k \mathbb E[e^{tX_1}]\\ &= \mathbb E[e^{tX_1}]^k \\ &= (1-p + pe^t)^k. \end{align} Set $$p=\frac29$$ and we have your function $$M(t)$$. Then $$\mathbb P(X=3) =\binom k 3 \left(\frac29\right)^3\left(\frac79\right)^{k-3}= \frac8{343}\binom k 3 \left(\frac79\right)^k.$$ To determine the factorial moments of $$X$$, we instead use the probability generating function $$P(s) = \mathbb E[s^X]$$. Here $$P(s) = \left(\frac79 + \frac29s\right)^k$$, and the $$r^{\mathrm{th}}$$ factorial moment is given by \begin{align} \lim_{s\to1^-} \frac{\mathsf d^r}{\mathsf ds^r} P(s) &= \lim_{s\to1^-} \frac{\mathsf d^r}{\mathsf ds^r} \left(\frac79 + \frac29s\right)^k\\ &= \left(\frac29\right)^r\left(\frac79 + \frac29s\right)^{k-r}\frac{k!}{(k-r)!}, \ r\leqslant k, \end{align} and zero for $$r>k$$.