# The Poisson distribution and log-linear models

First, a video.

This week you will:

Imagine sitting on the banks of the river Seine in Paris, fishing. Fish swim past randomly at a constant rate. If we catch fish for an hour, how many fish do we catch? Because they swim past randomly, the number will vary. it will also depend on the density of fish (and how fast they swim etc. etc.). But if we catch them at rate $$\mu$$, the mean number we catch in time $$t$$ will be $$\lambda = \mu t$$. The actual number will vary, and will follow a Poisson distribution:
$Pr(N=r | \lambda) = \frac{\lambda^r e^{-\lambda}}{r!}$