Question

Let x and y be continuous random variables with joint density function: fx,y (x, y) = ( 2(x y) for 0 < x < y < 1 0 otherwise. Calculate E[y].

Answer 1

E[y], the expected value of y, can be found by setting up and evaluating the double integral E[y] = ∫_0^1 ∫_x^1 2(xy)y dy dx over the given range of x and y.

Step-by-step explanation:

The student asked how to calculate E[y], the expected value of a continuous random variable y, given a joint density function fx,y (x, y) = 2(xy), for the range of 0 < x < y < 1.

To find E[y], integrate y multiplied by the joint density over the range of x and y. Set up the integral as follows:

E[y] = ∫_0^1 ∫_x^1 2(xy)y dy dx.

First, integrate with respect to y from x to 1, then integrate the result with respect to x from 0 to 1.

This will give you E[y], which represents the average value of y over its range of potential values given the probability density function.