Question

Suppose that x and y are jointly continuous with joint probability density function f(x, y) = ( 2e -(x 2y) for x > 0, y > 0 0 else. Show that x and y are independent random variables and provide their marginal distributions.

Answer 1

To show that x and y are independent random variables, we need to check if their joint probability density function can be factored into the product of their marginal probability density functions. Given the joint probability density function f(x, y) = 2e^-(xy), we can find the marginal probability density functions of x and y by integrating the joint probability density function. By showing that the joint probability density function can be factored into the product of the marginal probability density functions, we can conclude that x and y are independent random variables.

Step-by-step explanation:

To show that x and y are independent random variables, we need to check if their joint probability density function can be factored into the product of their marginal probability density functions.

Given that the joint probability density function is f(x, y) = 2e^-(xy) for x > 0, y > 0, and 0 elsewhere, we can find the marginal probability density functions by integrating the joint probability density function.

The marginal probability density function of x, f(x), is obtained by integrating f(x, y) with respect to y:

f(x) = ∫(0 to ∞) 2e^-(xy) dy = -2e^-(xy) / x | (0 to ∞) = 2/x for x > 0

The marginal probability density function of y, f(y), is obtained by integrating f(x, y) with respect to x:

f(y) = ∫(0 to ∞) 2e^-(xy) dx = -2e^-(xy) / y | (0 to ∞) = 2/y for y > 0

Since we have f(x) ≠ 0 and f(y) ≠ 0 for all x and y, and f(x, y) = f(x) * f(y), we can conclude that x and y are independent random variables.