Here's a description of the problem:
Suppose we have x apples and y oranges where each apple has a probability pa to rot and each orange has a probability po to rot, then what's the pdf of the total number of fruit z being rotten.
This pdf looks like a product of two binomial distributions on the surface. However, since z = x+y, x >= 0, y >= 0, then it is actually necessary to sum over all combinations of scenarios (i.e. 6 fruit rotten = 3 apples rotten + 3 oranges rotten or 6 fruits rotten = 1 apple rotten + 5 oranges rotten).
By intuition, I worked out the expectation to be x*pa + y *po and it appears to be correct when I manually tested my problem on a spreadsheet. However, I am not quite sure how that expectation can be derived from this messy pdf.
Sorry, I am a bit confused. How is this a sum of two binomials?
For example, if we want the probability of having 2 rotten fruits, then this probability is:
prob of having 2 rotten apples * prob of having 0 rotten oranges
prob of having 1 rotten apple * prob of having 1 rotten orange
prob of having 0 rotten apples * prob of having 2 rotten oranges
That doesn't really look like a sum of two binomials or maybe I am misunderstanding something.
I am confused...
The pdf of my rotten fruit distribution is the product of the pdf of the apple binomial pdf and the orange binomial pdf.
If it is instead a sum of the two pdfs, then wouldn't my rotten fruit distribution not be a well-formed distribution? For example, if there are one apple and one orange with their respective probability of rotting being both 1. Then the probability of having two rotten fruits would be 2 because we are summing their pdfs...
Sorry for sounding like an idiot, I am not very good with stats!