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

amhailim

amhailim

Answered question

2022-09-23

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

Answer & Explanation

belidla5a

belidla5a

Beginner2022-09-24Added 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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