Derivatives of PGF of Bernoulli Distribution

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:


 * $$\frac {d^k} {ds^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:
 * $$\frac d {ds} \Pi_X \left({s}\right) = p$$

Again, $$p$$ is constant, so from Derivative of Constant:
 * $$\frac d {ds} p = 0$$

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

Proof 2
We can directly use the result Derivatives of PGF of Binomial Distribution, setting $$n = 1$$.