Let X and Y be independent random variables each having the uniform density on {0,1,...,N}. Find P(X ge Y)

amhailim 2022-09-23 Answered
Let X and Y be independent random variables each having the uniform density on {0,1,...,N}. Find P ( X Y )
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Answers (1)

belidla5a
Answered 2022-09-24 Author has 8 answers
Using the law of the total probability (conditioning on X), we have that
P ( X Y ) = x = 0 N P ( X Y | X = x ) P ( X = x )
= x = 0 N P ( Y x ) P ( X = x )
= x = 0 N x + 1 N + 1 1 N + 1
= 1 ( N + 1 ) 2 x = 0 N x + 1
= ( N + 1 ) ( N + 2 ) 2 ( N + 1 ) 2 = N + 2 2 ( N + 1 )
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