Show that for a Geometric distribution, the probability generating function is given by $\frac{ps}{1-qs}$, $q=1-p$

Question

Suppose that $X$ has a geometric distribution with probability mass function $P(X=x) = q^{i-1}p$, $i=1,2,...$ and $q=1-p$

Show that its probability generating function is given by $ \pi(s)=\frac{ps}{1-qs}$. Hence show that $E(x)=\frac{1}{p}$ and $Var(X)=\frac{q}{p^2}$

Hi everyone, I am doing this question for exam practice, and I can't seem to get the correct answer. And to be honest, I am just working through it mechanically and don't have a great understanding of the probability generating functions.

Here is what I have:

$$\pi(s)=E(S^X)=\sum^\infty_{i=0}q^{i-1}p\cdot s^i$$ $$= p\sum^\infty_{i=0}q^{i-1}\cdot s^i=p\sum^\infty_{i=0}\frac{q^i}{q}\cdot s^i$$ $$=\frac{p}{q}\sum^\infty_{i=0}(qs)^i$$

Then using the sum of a geometric series formula, I get:

$$=\frac{p}{q}(\frac{1}{1-qs})$$

Now I am stuck. I feel like I am close, but am just missing something. I'll be ok with deriving the expected value and variance once I can get past this part.

As an addition I was wondering if anyone could also give me a bit of an 'idiots' explanation of the probability generating function, as I am struggling to understand it conceptually. $s$ seems to be the dependent variable, but my lecturer hasn't explained what exactly it is.

Many thanks in advance!

Answer 1

It's normal you'd arrive at the wrong answer in this case. The problem is that your index is wrong. There are two definitions for the pdf of a geometric distribution. The one you use, where $E(X)=\frac{1}{p}$ is defined from 1 to infinity. At zero it is not defined. So, the generating function needs to take this into account, as well.

$$\pi(s)=E(S^X)=\sum^\infty_{i=1}q^{i-1}ps^i$$ $$= ps\sum^\infty_{i=1}(qs)^{i-1}=ps\sum^\infty_{i=0}(qs)^i$$ $$=\frac{ps}{1-qs}$$

If you use the alternative definition, where $P(Y=y)=q^ip$, then the pdf is defined at zero. In this case the generating function converges to $\frac{p}{1-qs}$.

As for what $s$ represents, as far as I know it represents nothing. Generating functions are derived functions that hold information in their coefficients. They are sometimes left as an infinite sum, sometimes they have a closed form expression. Take a look at the wikipedia article, which give some examples of how they can be used. Here and here.wiki article probability generating functions and wiki article generating functions