Let X and Y be independent random variables having geometric densities with parameters p_(1) and p_(2) respectively. Find P(X ge Y)

sooxicyiy

sooxicyiy

Answered question

2022-09-06

Let X and Y be independent random variables having geometric densities with parameters p 1 and p 2 respectively. Find P ( X Y ).

Answer & Explanation

ivice7u

ivice7u

Beginner2022-09-07Added 18 answers

Using the law of the total probability (conditioning on X), we have that
P ( X Y ) = x = 0 P ( X Y | X = x ) P ( X = x )
= x = 0 P ( Y x ) P ( X = x )
= x = 0 ( 1 ( 1 p 2 ) x + 1 ) ( 1 p 1 ) x p 1
= x = 0 ( 1 p 1 ) x p 1 ( 1 p 2 ) x = 0 [ ( 1 p 1 ) ( 1 p 2 ) ] x
= 1 1 p 2 1 ( 1 p 1 ) ( 1 p 2 ) = p 2 p 1 + p 2 p 1 p 2

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