Question

Let x and y be two random variables with the joint pdf f (x, y) = {2, 0 < x < y < 1, 0, elsewhere. Find the following:

Answer 1

To find the probability that x<2, we integrate the joint PDF over the region where x is less than 2 and y is between x and 1.

Step-by-step explanation:

The joint probability density function (PDF) for random variables x and y is given as f(x,y) = 2, 0<x<y<1 and 0 elsewhere. To find the probability that x<2, we integrate the joint PDF over the region where x is less than 2 and y is between x and 1. This can be expressed as:

P(x<2) = ∫∫2 dxdy, where the integration is done over the region where 0<x<2 and x<y<1.

Integrating this expression, we get:

P(x<2) = ∫∫2 dy dx = ∫0 to 2 (∫x to 1 2 dy) dx = ∫0 to 2 (2(1-x)) dx = ∫0 to 2 (2-2x) dx = [2x-x^2] from 0 to 2 = 4-4 = 0

Therefore, the probability that x<2 is 0.