Probability Generating Function of Bernoulli Distribution

Theorem

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

Then the p.g.f. of $X$ is:

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

where $q = 1 - p$.

Proof

From the definition of p.g.f:

$\ds \map {\Pi_X} s = \sum_{x \mathop \ge 0} \map {p_X} x s^x$

From the definition of the Bernoulli distribution:

$\map {p_X} x = \begin{cases}

p & : x = a \\ 1 - p & : x = b \\ 0 & : x \notin \set {a, b} \\ \end{cases}$

So:

\(\ds \map {\Pi_X} s\)

\(=\)

\(\ds \map {p_X} 0 s^0 + \map {p_X} 1 s^1\)

\(\ds \)

\(=\)

\(\ds \paren {1 - p} + p s\)

Hence the result.

$\blacksquare$

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.2$: Integer-valued random variables: $(10)$