Question

Show that the Moment generating function for a Bernoulli (p) random variable X is given by

MX(t) = pet + q where q = 1 − p.

Answer 1

Solution:

From the definition of the Bernoulli distribution, X has probability mass function:

$\operatorname{Pr}(X=n)= \begin{cases}q & : n=0 \\ p & : n=1 \\ 0 & : n \notin\{0,1\}\end{cases}$

From the definition of a moment generating function:

$M_{X}(t)=\mathrm{E}\left(e^{t X}\right)=\sum_{n=0}^{1} \operatorname{Pr}(X=n) e^{t n}$

So:

$\begin{aligned} M_{X}(t) &=\operatorname{Pr}(X=0) e^{0}+\operatorname{Pr}(X=1) e^{t} \\ &=q+p e^{t} \end{aligned}$

where $q=1-p$

Hence, proved.