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November 3rd, 2017, 03:46 PM   #1
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Question Distribution of Function of Random Vector

N, K, and i are positive integers for the following:

Let the random vector X be represented by <X1, X2, ... , XN>, where X1, X2, ... XN are all random variables which follow a discrete uniform distribution with parameters 1 and K. Hence, P(Xi = xi) = 1/K on 1 <= xi <= K for 1 <= i <= N.

Let the random variable Y = f(X vector) = the number of distinct entries in X vector.

So, for example, if N = 3 and K = 9:
f(<1, 1, 1>) = 1
f(<1, 2, 1>) = 2
f(<1, 2, 3>) = 3
f(<3, 1, 3>) = 2
f(<9, 9, 9>) = 1
f(<1, 8, 8>) = 2
etc.

What is the distribution of Y?

My attempt so far:

I believe that X vector follows a Multinomial Distribution with parameters N and the correspondingly sized probability vector <1/K, ... , 1/K>. This may be helpful.

I also think that the support of Y is the set of all integers between 1 and N (inclusive).

But I get lost from this point; I'm still nowhere near finding the distribution of Y. Do you have any ideas?
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November 3rd, 2017, 11:36 PM   #2
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Figured it out after writing a bit of python code:
http://prntscr.com/h61pen

Crazy stuff.
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