FS1 - Probability Generating Functions - Geometric

In this section we will look at properties of the Geometric distribution using its probability generating function.

Let X \sim \mathrm {Geo}(p). Recall that:

\mathrm P(X = n) = p (1 - p)^{n - 1}

for integer n \ge 1.

It helps to remember that:

\displaystyle \sum_{n = 0}^\infty x^n

converges to \dfrac 1 {1 - x} if |x| < 1, and diverges otherwise.

With that said, we have that G_X, the probability generating function of X is given by:

\begin{align*}G_X(t) & = \sum_{n = 0}^\infty \mathrm P(X = n) t^n \\ & = \sum_{n = 1}^\infty p(1 - p)^{n - 1} t^n \\ & = pt \sum_{n = 1}^\infty ((1 - p)t)^{n - 1} \\ & = pt \sum_{n = 0}^\infty ((1 - p)t)^n\end{align*}

We can see now that the probability generating function of X exists only when |(1 - p)t| < 1. That is, when |t| < \dfrac 1 {1 - p}.

This gives us kind of a “sanity check” - if you get to the point of substituting a t into G_X (for example - if you confused p and q = 1 - p when processing the context of a problem) with |t| \ge \dfrac 1 {1 - p}, you know you’ve made a mistake and will get a nonsense answer in return.

If |t| < \dfrac 1 {1 - p}, we then have:

\displaystyle G_X(t) = \frac {pt} {1 - (1 - p)t}

Setting q = 1 - p, you can write this as:

\displaystyle G_X(t) = \frac {pt} {1 - qt}

for neatness, we will use this form.

We will now, as before, find \mathrm E(X) and \mathrm {var}(X) via repeated differentiation, using the quotient rule.

We have:

\begin{align*}G'_X(t) &= \frac {p(1 - qt) - pt(-q)} {(1 - qt)^2} \\ & = \frac {p - pqt + pqt} {(1 - qt)^2} \\ & = \frac p {(1 - qt)^2}\end{align*}

setting t = 1 we get:

\begin{align*}\mathrm E(X) & = G'_X(1) \\ & = \frac p {(1 - q)^2} \\ & = \frac p {p^2} \\ & = \frac 1 p\end{align*}

Differentiating again using the chain rule we have:

\begin{align*}G''_X(t) & = -\frac {2p(-q)} {(1 - qt)^3} \\ & = \frac {2pq} {(1 - qt)^3}\end{align*}

setting t = 1 gives:

\begin{align*}G''_X(1) & = \frac {2pq} {(1 - q)^3} \\ & = \frac {2pq} {p^3} \\ & = \frac {2q} {p^2}\end{align*}

putting the previous two results together:

\begin{align*}\mathrm {var}(X) & = G'_X(1) + G''_X(1) - (G'_X(1))^2 \\ & = \frac 1 p + \frac {2q} {p^2} - \frac 1 {p^2} \\ & = \frac p {p^2} + \frac {2 - 2p} {p^2} - \frac 1 {p^2} \\ & = \frac {1 - p} {p^2}\end{align*}

As we expected. We can also write this as:

\displaystyle \mathrm{var}(X) = \frac q {p^2}

Summary

Let X \sim \mathrm {Geo}(p). The probability generating function, G_X of X is given by:

\displaystyle G_X(t) = \frac {pt} {1 - (1 - p)t} = \frac {pt} {1 - qt}

valid for |t| < \dfrac 1 {1 - p} = \dfrac 1 q. (with q = 1 - p)

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