Question

In: Statistics and Probability

Let X and Y be i.i.d. geometric random variables with parameter (probability of success) p, 0<p<1....

Let X and Y be i.i.d. geometric random variables with parameter (probability of success) p, 0<p<1.

(a) Find P(X>Y).

(b) Find P(X+Y=n) and P(X=k|X+Y=n), for n=2,3,..., and k=1,2,...,n−1.

I need an answer asap please. Thank you.

Expert Solution

a)

P(X > Y) + P(X < Y) +P(X = Y) = 1

since X and Y are iid with same parameter

P(X > Y)= P(X <Y)

hence

2 P(X > Y) = 1 - P(X = Y)

now

= p/(2-p)

hence

P(X > Y) = 1/2 * ( 1- p/(2-p))

= 1/2 * (2-2p)/(2-p)

= (1-p)/(2-p)

b)

P(X = k | X+Y = n)

=( P(X = k) and P(X + Y = n)) / P(X + Y = n)

= P(X = k) * P(Y = (n-k)) / P(X + Y = n)

= p(1-p)^(k-1) * p * (1-p)^(n-k-1) / ( (n-1)(1-p)^(n-2) p^2)

= 1/(n-1)

orchestra answered 2 years ago

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