Lecture 12

Today's lecture was on probability generating functions. A very nice tripos question is the following, 2004/2/II/9F. You should now be able to do this (and could discuss in your next supervision)

A non-standard pair of dice is a pair of six-sided unbiased dice whose faces are numbered with strictly positive integers in a non-standard way (for example, (2,2,2,3,5,7) and (1,1,5,6,7,8)). Show that there exists a non-standard pair of dice A and B such that when thrown

P(total shown by A and B is $n$) = P(total shown by pair of ordinary dice is $n$).

for all $2\leq n\leq 12$,

[Hint: $(x+x^2+x^3+x^4+x^5+x^6)=x(1+x)(1+x^2+x^4)=x(1+x+x^2)(1+x^3)$.]

We saw today that generating functions are useful in realms beyond probability, as example being that

"the $n$th Fibonacci number, $F_n$, is the coefficient of $x^n$ in the expansion of the
function $1/(1−x−x^2)$ as a power series about the origin."

That is a quote from chapter 1 of Herbert Wilf's fun book called generatingfunctionology, whose second edition is free to download. In his book you can read about many weird and wonderful things that can be done with generating functions. He starts Chapter 1 by writing

"A generating function is a clothesline on which we hang up a sequence of numbers for display."

Generating functions can encapsulate complicated things in a lovely way.  For example, the coefficient of $x^n/n!$ in the power series expansion of $e^{e^x-1}$ is the Bell number $B(n)$, i.e. the number of different partitions that can be made from a set of $n$ elements. In Chapter 1 of Wilf's book you can see a derivation of this fact.