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November 3rd, 2017, 02:46 PM  #1 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0  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? 
November 3rd, 2017, 10:36 PM  #2 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0  

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distribution, function, random, vector 
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