Show that the Moment generating function for a Bernoulli (p) random variable X is given by
MX(t) = pet + q where q = 1 − p.
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}\n\nM_{X}(t) &=\\operatorname{Pr}(X=0) e^{0}+\\operatorname{Pr}(X=1) e^{t} \\\\\n\n&=q+p e^{t}\n\n\\end{aligned}"
where "q=1-p"
Hence, proved.
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