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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.

Any suggestions?

Thanks.

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# Product of two binomials

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