Question

suppose that x and y have a continuous joint distribu-tion for which the joint p.d.f. is as follows: f(x, y) = 12y2 for 0 ≤ y ≤ x ≤ 1, 0 otherwise. find the value of e(xy).

Answer 1

To find the value of E(XY), we need to calculate the expected value of the product of x and y. Follow these steps: Determine the limits of integration, set up the integral, and perform the double integration.

Step-by-step explanation:

The joint p.d.f. of x and y is given as f(x, y) = 12y^2 for 0 ≤ y ≤ x ≤ 1, 0 otherwise. To find the value of E(XY), we need to calculate the expected value of the product of x and y.

Step 1: Determine the limits of integration

The limits of integration are given as 0 ≤ y ≤ x ≤ 1. We will integrate with respect to y first, and then with respect to x.

Step 2: Set up the integral

The integral expression for E(XY) is ∫∫ f(x, y) * xy dy dx, with the limits of integration as 0 ≤ y ≤ x ≤ 1.

Step 3: Perform the double integration

Integrating the given expression, we get E(XY) = ∫ ∫ (12y^2)(xy) dy dx, with the limits of integration as 0 ≤ y ≤ x ≤ 1. Solving the integral will give us the value of E(XY).