Travis: Segment 41

From Computational Statistics (CSE383M and CS395T)
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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?