Question

The joint probability density function of X and Y is given by f(x,y) = 2e−(x+2y) 0≤x<[infinity], 0≤y<[infinity] .Find P{X < Y}

Answer 1

To find the probability that X < Y, we need to integrate the joint density function over the region where X < Y. This region is the upper triangular half of the xy-plane, where x is between 0 and infinity, and y is between x and infinity. The integral is:

PX<Y=int0infty​intxinfty​2e−(x+2y)dydx

Using integration by parts, we can simplify this integral as follows:

PX<Y=int0infty​left[−e−(x+2y)right]xinfty​dx=int0infty​e−3xdx=left[−frac13e−3xright]0infty​=frac13

Therefore, the probability that X < Y is 1/3.

Explanation: