Derivatives of PGF of Bernoulli Distribution/Proof 2

Theorem

Let $X$ be a discrete random variable with the Bernoulli distribution with parameter $p$.

Then the derivatives of the PGF of $X$ with respect to $s$ are:

$\dfrac {\d^k} {\d s^k} \map {\Pi_X} s = \begin{cases} p & : k = 1 \\ 0 & : k > 1 \end{cases}$

Proof

Follows directly from Derivatives of PGF of Binomial Distribution, setting $n = 1$.

$\blacksquare$