Given a set $S$, where each member $s_i$ has a probability to be sampled $p_i$. What is the expected number of unique samples when randomly sampling $n$ items with replacement?
My question is similar to this question with one important difference. In my case the probability of sampling the different items of the set is not uniform. Each item has a unique probability to be sampled that will be constant over all samples.
Aftwe looking at Mike's excellent answer to this question, I was able to produce the following answer:
Introduce an indicator variable $X_i$ that is 1 if $s_i$ was sampled from the set. Then we get the number of unique items as: $$Y = X_1+X_2+\cdots+X_m$$ where m is the size of S.
By linearity of expectation, $$E[Y] = E[X_1] + E[X_2] + \cdots + E[X_m].$$
So now the problem reduces to calculating the $E[X_i]$ for each $i$. But this is fairly easy, as $$E[X_i] = 1-(1-p_i)^n$$
Therefore, $$E[Y] = \sum_{i=0}^{m} 1-(1-p_i)^n $$