Derivatives of PGF of Bernoulli Distribution

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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 1

The Probability Generating Function of Bernoulli Distribution is:

$\Pi_X \left({s}\right) = q + ps$

where $q = 1 - p$.


We have that for a given Bernoulli distribution, $p$ and $q$ are constant.

So, from Derivative of Constant, Sum Rule for Derivatives, Derivative of Identity Function and Derivative of Constant Multiple:

$\dfrac {\mathrm d} {\mathrm d s} \Pi_X \left({s}\right) = p$


Again, $p$ is constant, so from Derivative of Constant:

$\dfrac {\mathrm d} {\mathrm d s} p = 0$


Higher derivatives are also of course zero, also from Derivative of Constant.

$\blacksquare$


Proof 2

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

$\blacksquare$