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

tamolam8 2022-09-13 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)

shosautesseleol
Answered 2022-09-14 Author has 16 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 1 N + 1 1 N + 1
= 1 ( N + 1 ) 2 x = 0 N 1
= ( N + 1 ) ( N + 1 ) 2 = 1 N + 1

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