Expected number of distinct integer values I have a process that generates integer values following a binomial distribution (n,p). I would like to know what the expected number of distinct integer values given T trials is. Can someone provide the relevant theory?
For instance, let's say our distribution's mean is 450 with a variance of 50 and we request 10 trials. We might get: 450, 450, 451, 449, 450, 460, 425, 445, 448, 450; i.e. 7 distinct integers. What is the expected value of this statistic?
 A: If you always have such a small standard deviation as $\sigma=\sqrt{5},$ then there are
relatively few likely integers even with $n$ larger than 10 samples. Maybe a
dozen or so within $3\sigma$ of $\mu.$ So the expected number $E(U)$ of unique
values will be relatively small.
Trivial simulations gave $E(U) \approx 6.05$ with $n = 10$, $E(U) \approx 8.19$ with 20, and $E(U) \approx 10.34$ with 50; all with $SD(U)$ a little over $1.$
Sample simulation in R:
u = replicate( 10^6, length(unique(round(rnorm(10, 450, sqrt(5))))) )
mean(u); sd(u)
## 6.052771
## 1.058554

With very much larger standard deviations, you might be able to view ties as rare
events and get a serviceable Poisson model for $U.$
Note: This is vaguely reminiscent of the famous birthday problem except that
there are fewer possible values (than 365) and not all values are
equally likely (more ties near $\mu$). Also, discussions of the birthday
problem usually center on probability of a match, not average numbers
of matches.
A: You can solve this using dynamic programming.  For each node representing "a set of integers that you have visited after i sampled variates", you keep track of the total probability of reaching it.  After you populate this tree for all reached nodes having $i\le n$, you can calculate the expected value of distinct integers by taking a weighted sum over nodes with $i=n$.
Two optimizations are making use of the symmetry to reduce the number of nodes you have to populate and trimming nodes whose probability of being reached is less than the precision of your desired result.
