[CST-2] CSM 96/9/3

[Tim Harris]

> Can someone kindly explain the last part of Tim Harris' solution to 96/9/3
> "As an extension, you could try to confirm the validity of the assertion
> that a value from Er can be generated by summing r values from E1 etc.
> ..." Example class 1

This last part is meant very much as an extension rather than as something
that is part of the answer to the question.

The question refers to two different procedures for generating random
variates from an r-stage Erlangian distribution.  The first (the one
to use in practise!) is to sum r independent identically-distributed
values from an exponential distribution.  The second is to take the 
pdf given in the question and to follow the usual steps of integrating
that to get the cdf, inverting that and using it to map uniformly
distributed values directly to ones from the required Erlangian
distribution.

This raises the question of why both of these schemes are valid and
yield values with the same distribution: i.e. where the pdf given
in the question comes from.

The suggestion at the end is to prove this by induction.  For the
base case, E_1, all that is needed is to show the the pdf given in
the question matches that of the exponential when r is set to 1.
For the inductive step, consider expressing E_r as the sum of a
value with E_{r-1} distribution and an additional value from E_1.

Tim