I'm trying to come up with a reasonable way to determine the weights in a mixture distribution. Let us consider the following example:
There are two districts ($i=1,2$) in a city, both of which share one hospital. The distance between the districts and the hospital is denoted by $D_i, i=1,2.$
Incidents occur in both districts according to a Poisson process with intensity $\lambda$ (the same parameter for both districts).
However, district 1 can transfer patients to the hospital at a unique rate of $\theta$. In other words, transportation opportunities become available at rate $\theta$ and when it becomes available, it takes all patients waiting in that district to the hospital. Obviously, if there are not patients, there will be no dispatch and let us assume the vehicle is big enough to fit everyone!
For district 2 however, there's only one vehicle that is always available but it must be full before it can be dispatched and let us denote the capacity of this vehicle by $\gamma$
If we denote the number of patients leaving every district by $X_i$, for district 1, this is a random variable that follows a geometric distribution so we have $\mathbb{E}[X_1] = \frac{\lambda}{\theta}$.
For district 2, it is just a constant number that is $\mathbb{E}[X_2] = \gamma.$
To calculate $\mathbb{E}[Y]$, that is the average number of patients arriving at the hospital at a given point in time, by definition, we have $$\mathbb{E}[Y] = \sum_{i=1,2} w_i \mathbb{E}[X_i].$$
Now what would be a neat way to determine $w_i, i = 1,2$? I think $$\text{probability}(Y = X_i) \propto \text{Frequency of dispatches from district } i,$$ which in turn depends on $\lambda$ and $\theta$ and $$\text{probability}(Y = X_i) \propto 1/D_i.$$
Any suggestions would be much appreciated.
Best Answer
As clarified in the comments above, we seek to find the average number of patients in a randomly-selected vehicle arriving at the hospital. As is correctly stated in the question, such a vehicle will either be from District 1 or District 2 with average numbers of patients $\lambda/\theta$ and $\gamma$, respectively.
The probabilities (i.e. the weights) for finding a vehicle from each District are simply the mean dispatch frequencies from the two districts, normalized to sum to 1: $$ P(\text{Vehicle is from District 1}) \;=\; \frac{\theta}{\theta + (\lambda/\gamma)} $$ $$ P(\text{Vehicle is from District 2}) \;=\; \frac{\lambda/\gamma}{\theta + (\lambda/\gamma)} $$
The distances of each district from the hospital have no bearing, since - assuming the system is at equilibrium - every vehicle dispatched to the hospital arrives at the hospital, so the pipeline to the hospital will be continuously populated by previously-dispatched vehicles.
Another way to look at it: If the arrival frequency were greater than the dispatch frequency, it would mean extra vehicles were being conjured into existence en route. If the arrival frequency were less than the dispatch frequency, it would mean vehicles were building up in the pipeline.