Derivatives of PGF of Bernoulli Distribution

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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 1

The Probability Generating Function of Bernoulli Distribution is:

$\map {\Pi_X} s = q + p s$

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:

$\dfrac {\d} {\d s} \map {\Pi_X} s = p$

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

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

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


Proof 2

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