# Travis: Segment 41

#### 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.)**

<math> P [(N(t+ \tau) - N(t)) = k] = \frac{e^{-\lambda \tau} (\lambda \tau)^k}{k!} \qquad k= 0,1,\ldots </math> where k is the number of events between <math> t</math> and <math> t + \tau</math>. So if we plug in <math>k = 1 </math> we get

<math> p(k=1) =(\lambda \tau) e^{-\lambda \tau} </math> which is the exponential distribution with parameter <math> \lambda \tau </math>

**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 <math>k_1</math> or <math>k_2</math> 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 <math>t_i</math> 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 <math>N</math> of data points <math>t_i</math> that you need to clearly distinguish between these hypotheses?**