How to obtain probability distribution from the generating function $G(s) = e^{a(s-1)^2}$?

Question

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this:

$G(s) = e^{a(s-1)^2}=\sum s^np(n)$

I need first to do Maclaurin expansion of the exponential and then get the $n$th order term for $p(n)$.

My first thinking was it would be simple to calculate the derivatives. But it turns out to be much more difficult and also very interesting to generalize the $n$ order derivative.

I list a table for the powers and coefficients of each derivative order, finding that the powers are odd numbers for odd $n$, even numbers for even $n$, the coefficients are associated to $(n-1)$th order's powers and coefficients. It is easy to see the association but I cannot generalize it.

This calculation is also like $z$ transform but no existing result for it.

Anyone could give a shot and help me out?

Answer 1

$G$, using your notation, is known in probability as the probability generating function. See here for more details. It is seen on this link that $$p(k) = \dfrac{\left.G^{(k)}(s)\right|_{s=0}}{k!}$$ where $G^{(k)}$ denotes the $k$th derivative.

Answer 2

Using Maple and the information given at http://en.wikipedia.org/wiki/Hermite_distribution

it is possible to obtain

The first ten probabilities are

These results can be obtained as follows.

The generating function for the Hermite polynomials is

$${{\rm e}^{2\,xt-{t}^{2}}}=\sum _{n=0}^{\infty }{\frac {H_{{n}} \left( x \right) {t}^{n}}{n!}}$$

Then

$${{\rm e}^{-\alpha\, \left( 1-s \right) ^{2}}}={{\rm e}^{-\alpha}}{ {\rm e}^{2\,s\alpha-\alpha\,{s}^{2}}} $$

making the change $$t=\sqrt {\alpha}s$$ we have

$${{\rm e}^{-\alpha\, \left( 1-s \right) ^{2}}}={{\rm e}^{-\alpha}}{ {\rm e}^{2\,t\sqrt {\alpha}-{t}^{2}}} $$

and then we obtain

$${{\rm e}^{-\alpha}}{{\rm e}^{2\,t\sqrt {\alpha}-{t}^{2}}}={{\rm e}^{- \alpha}}\sum _{n=0}^{\infty }{\frac {H_{{n}} \left( \sqrt {\alpha} \right) {t}^{n}}{n!}}$$

it is to say

$${{\rm e}^{-\alpha\, \left( 1-s \right) ^{2}}}={{\rm e}^{-\alpha}}\sum _{n=0}^{\infty }{\frac {H_{{n}} \left( \sqrt {\alpha} \right) \left( \sqrt {\alpha}s \right) ^{n}}{n!}}$$

Then we write

$$ \sum _{n=0}^{\infty }p_{{n}}{s}^{n}={{\rm e}^{-\alpha}}\sum _{n=0}^{ \infty }{\frac {H_{{n}} \left( \sqrt {\alpha} \right) \left( \sqrt { \alpha}s \right) ^{n}}{n!}}$$

Finally we obtain

$$p_{{n}}={\frac {{{\rm e}^{-\alpha}}H_{{n}} \left( \sqrt {\alpha} \right) \left( \sqrt {\alpha} \right) ^{n}}{n!}}$$