Question

Determine the probability that the difference between random variables Y and X is greater than 3, expressed as a fraction. The joint probability density function of X and Y is given by:

f(x, y) = {
k(x + y), 0 < x < 2, 1 < y < 4
0, otherwise

Answer 1

To determine the probability that the difference between random variables Y and X is greater than 3, integrate the given joint probability density function.

Step-by-step explanation:

To determine the probability that the difference between random variables Y and X is greater than 3, we need to integrate the given joint probability density function. The joint probability density function is given by:

f(x, y) = { k(x + y), 0 < x < 2, 1 < y < 4
0, otherwise

We first need to find the value of the constant k. To do so, we can integrate the joint probability density function over its entire domain and set it equal to 1:

1 = ∫14 ∫02 k(x + y) dx dy

Solving this integral, we find that k = 1/16. Now, we can calculate the probability:

P(Y - X > 3) = ∫14 ∫x + 32 (1/16)(x + y) dx dy

Calculating this double integral will give us the probability.