Derivatives of PGF of Bernoulli Distribution/Proof 2

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Theorem

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

Then the derivatives of the PGF of $X$ w.r.t. $s$ are:

$\dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({s}\right) = \begin{cases} p & : k = 1 \\ 0 & : k > 1 \end{cases}$


Proof

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

$\blacksquare$