# Eleisha's Segment 41: Markov Chain Monte Carlo, Example 2

**To Calculate**

1. Show that the waiting times (times between events) in a Poisson process are Exponentially distributed. (I think we've done this before.)

2. Plot the pdf's of the waiting times between (a) every other Poisson event, and (b) every Poisson event at half the rate.

3. Show, using characteristic functions, that the waiting times between every Nth event in a Poisson process is Gamma distributed. (I think we've also done one before, but it is newly relevant in this segment.)

**To Think About**

1. In slide 5, showing the results of the MCMC, how can we be sure (or, how can we gather quantitative evidence) that there won't be another discrete change in k_1 or k_2 if we keep running the model longer. That is, how can we measure convergence of the model?

2. Suppose you have two hypotheses: H1 is that a set of times t_i are being generated as every 26th event from a Poisson process with rate 26. H2 is that they are every 27th event from a Poisson process with rate 27. (The mean rate is thus the same in both cases.) How would you estimate the number N of data points t_i that you need to clearly distinguish between these hypotheses?

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