Question

Let the random variable Y denote the time (minutes) for which a customer is waiting for the beginning of a service in a service station since its arrival, and let X denote the time (minutes) until the service is completed since its arrival at the service station. Since both X and Y measure the time since the arrival of the customer at the service station, always Y < X is true. The joint probability density function for X and Y is given as follows: fxy(x,y) = C (x + y), for 0 < x < 2 and 0

Answer 1

The constant C is determined by the condition that the total probability for all possible outcomes must equal to 1. To determine the value of C, we use the double integral method to evaluate the total probability.

We will evaluate the integral of fxy(x, y) over all possible values of x and y. Since 0 < y < x < 2, The integral will run from 0 to 2 for x, and from 0 to x for y.

Start by taking the integral of the given function with respect to y, which is Cyx + 1/2 * C * y^2, evaluated from y=0 to y=x.

This gives C*x^2 + C/2 * x^2 = C/2 * (3x^2), for 0 < x < 2.

Now take the integral of C/2 * (3x^2) with respect to x, over the interval 0 to 2.

That gives C/2 * [x^3] evaluated from 0 to 2, which equals C/2 * [8] = 4C.

Since the total probability must equal 1, we set the above integrated function to be equal to 1 and solve for C, we get C = 1/4.

So, the joint probability density function for X and Y is fxy(x, y) = 1/4 * (x + y), for 0 < y < x < 2.