Bayesian inference of Poisson likelihood and exponential prior.

[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

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.

 

[lippyka]

The parameter \theta represents the expected number of calls arriving at a switchboard in a given time interval. \lambda represents the rate of the Exponential distribution which is related to the average length of a call. To estimate \lambda , you would need to use the data in the table to calculate the average length of a call. For example, you could use the formula \lambda = \frac{t_i}{y_i} where t_i is the total call duration in minutes and y_i is the number of calls for each switchboard. Once you have estimated \lambda , you can then use it to calculate the posterior distribution for each switchboard.