Addition of 2 distributions

Consider a situation where we want to model the evolution of a count variable over time. For example, the number of patients in a hospital waiting room, the number of unfilled orders posted on a stock exchange, the number of fi…rms active in a particular economic market, the number of active life insurance policies a company has etc. To be specific, let Y_t denote the number of patients in a hospital waiting room at time t. Consider what happens as we move forward in time to t+1 (eg. move forward by …fteen minutes say). During that time, some patients will be seen by doctors and leave the queue, while other new patients will arrive. The general model for this looks like

Y*t = g (Y*(t-1)) + U_t

Where g (Y*(t-1) is some function (to be speci…fied) that measures how many of those waiting at time t +1 1 are still waiting at time t, and U_t represents the new arrivals to the waiting room between times t-1 and t. A classic model for this situation assumes that, conditional on Y*(t-1), g (Y*(t-1)) has a binomial distribution with parameters Y*(t-1) and , and that U*t has a Poisson distribution with parameter , and that g (Y*(t-1)) and U*t are independent of each other. This specifi…cation for g (Y*(t-1)) is motivated as follows. We imagine that for each of the Y*(t-1) individuals in the queue at time t - 1, an independent Bernoulli experiment is carried out — with probability the individual stays in the queue through to time t, and with probability (1 - a ) the individual leaves the queue between times t-1 and t. That is, we could write g (Y*(t-1)) =SUM(X*i from i=1 to Y*(t-1)) where X*i is a Bernoulli random variable that takes the value 1 with probability . Conditional on Y*(t-1), derive expressions for

A) Pmf of Y_t

B)The mean of Y_t

C)The variance of Y_t

Re: Addition of 2 distributions

Hey wilfhawk.

Can you show us the model that you have chosen to use for this problem? In general, your model could be as complicated as you wanted with all kinds of joint distributions taking any pieces of information it wants to.

You have written the model g (Y(t-1)) =SUM(Xi from i=1 to Y(t-1)). Is this the one you want to use?

If so, you should probably look at a compound Poisson process as well as probability generating functions for Bernoulli processes. You might also want to have a look at generalized linear models and link functions for your g().

I'd check out compound poisson processes first though. Have you covered them in your course?