## pure birth process

The Riddler has a rather simplistic riddle this week since it essentially asked for the expectation of a pure birth process (also known as the Yule process) at time t. Since the population size at time *t* has a geometric distribution with expectation

*e ^{λt}*.

It however took me a while to recover this result on my own on Easter afternoon, as I went for the integrals rather than the distribution itself and the associated differential equations. Interestingly (in a local sense!), I first following the wrong path of looking at the average time to the first birth, 1/*λ*, then to the second, 2/*λ*, and so on. Wrong since of course expectations do not carry this way… For a unit rate, *λ*=1, the average time to reach 10 births is about 3, while the average number of births over t=3 is essentially 20.

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