Wednesday, 11 February 2015

Lecture 12

Let me remind you that you can install copies of Mathematica and Matlab software. It can be fun to use these to help with an appreciation of things we are doing. Today I used this Mathematica program to calculate some probability generating functions.

(* Roll a fair die 10 times *) 
(* What is the probability the sum is 50? *)

p[z_] = (1/6) (z + z^2 + z^3 + z^4 + z^5 + z^6)
SeriesCoefficient[p[z]^10, {z, 0, 50}]
% // N

Series[1/(1 - x - x^2), {x, 0, 10}]

Table[Fibonacci[n], {n, 0, 10}]

Series[(1 - Sqrt[1 - 4 x])/(2 x), {x, 0, 10}]

Table[CatalanNumber[n], {n, 10}]

Mathematica is available here 
Matlab is here

A very nice tripos question on probability generating functions 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.