# Bayesian inference of Poisson likelihood and exponential prior.

1. Oct 26, 2011

### Inferior89

Hey I have some problems understanding my statistics homework.
I am given a data set giving the number of calls arriving to different switchboards in
three hours as well as the total phone call duration in minutes for each switch board.

Something like
Code (Text):
i    y_i    t_i
--------------
1    4      92
2    1      15
.     .         .
.     .         .
and so on.

Now we assume the likelihood for the number of calls is a Poisson distribution

$f(y | \theta) = \frac{\exp(- \theta) \theta^y }{y!}$

and an exponential prior for $\theta \sim \text{Exp}(\lambda)$

The posterior is thus a $\Gamma(1 + y, 1 + \lambda)$ where the second parameter is the rate.
As can be seen here http://en.wikipedia.org/wiki/Conjugate_prior#Discrete_likelihood_distributions

I have two questions.

* Nowhere in the assignment does it say what the parameter $\theta$ represents. Is it the average length of a call or maybe the waiting time between calls?

* How am I supposed to estimate $\lambda$ from the data in the table? The posteriors I got when using $1/\lambda = (180-t_i)/y_i$` to estimate the average waiting time for the two table values above can be seen here: http://i.imgur.com/dA9s7.png. However these distributions seem centered around the y_i values and not estimate the average waiting time.