Suppose thatthe joint density of X and Y is given by
(x,y)=e-(x/y).e(-y) ,0<x<�‡,0<y<�‡ and is o otherwise. Find P(X>1 / Y=y).
To find P(X > 1 | Y = y), we need to find the probability that X is greater than 1 given that Y is equal to y. We can calculate this using conditional probability.
The joint density function is given by f(x, y) = e^(-(x/y)) * e^(-y) for 0 < x < ∞ and 0 < y < ∞, and it is 0 otherwise.
To find P(X > 1 | Y = y), we need to compute the conditional probability using the joint density. The conditional probability formula is:
P(A | B) = P(A ∩ B) / P(B)
In this case, A = X > 1 and B = Y = y. Let's go step by step.
Step 1: Calculate the joint probability P(X > 1, Y = y) = ∫(from 1 to ∞)∫(from y to ∞) f(x, y) dx dy
= ∫(from 1 to ∞) e^(-x/y) * e^(-y) dx
= -e^(-x/y) * e^(-y) | (from 1 to ∞)
= -(0 - e^(-1/y) * e^(-y))
= e^(-1/y) * e^(-y) - 0
= e^(-1/y - y)
Step 2: Calculate the marginal probability P(Y = y) = ∫(from 0 to ∞) f(x, y) dx
= ∫(from 0 to ∞) e^(-x/y) * e^(-y) dx
= -e^(-x/y) * e^(-y) | (from 0 to ∞)
= -(0 - e^(0/y) * e^(-y))
= e^(0/y) * e^(-y) - 0
= e^(-y)
Step 3: Calculate the conditional probability P(X > 1 | Y = y) = P(X > 1, Y = y) / P(Y = y)
= e^(-1/y - y) / e^(-y)
= e^(-1/y)
Therefore, P(X > 1 | Y = y) = e^(-1/y).
So, the probability that X is greater than 1 given that Y is equal to y is e^(-1/y).