Question

Let X and Y be continuous random variables with the joint probability density function:

f(X, Y) = 2e^(-x) e^(-2y), for 0 < x < [infinity] and 0 < y < [infinity], otherwise.

Find P(2X < Y < X).

Answer 1

To find the probability P(2X < Y < X), we need to evaluate the integral of the joint probability density function over the given range. After simplifying the integral and solving for the inner and outer integrals, we find that the probability is equal to 1.

Step-by-step explanation:

To find the probability P(2X < Y < X), we need to find the area under the joint probability density function in the given range. The given joint probability density function is f(X, Y) = 2e^(-x) e^(-2y), for 0 < x < [infinity] and 0 < y < [infinity].

First, let's set up the integral to calculate the probability:

P(2X < Y < X) = ∫∫ 2e^(-x) e^(-2y) dy dx

Now, let's evaluate the integral:

P(2X < Y < X) = ∫ [0, ∞] ∫ [2x, ∞] 2e^(-x) e^(-2y) dy dx

Simplifying the integral:

P(2X < Y < X) = 2∫ [0, ∞] e^(-x) ∫ [2x, ∞] e^(-2y) dy dx

Now, solve the inner integral:

P(2X < Y < X) = 2∫ [0, ∞] e^(-x) [-1/2e^(-2y)] [2x, ∞] dx

P(2X < Y < X) = ∫ [0, ∞] e^(-x) (1 - 1/e^(-4x)) dx

Now, solve the outer integral:

P(2X < Y < X) = [-e^(-x) + 1/4e^(-4x)] [0, ∞]

P(2X < Y < X) = [0 - (-1)]

P(2X < Y < X) = 1