The management at a fast-food outlet is interested in the joint behavior of the random variables Y1, defined as the total time between a customer’s arrival at the store and departure from the service window, and Y2, the time a customer waits in line before reaching the service window. Because Y1 includes the time a customer waits in line, we must have Y1 ≥ Y2. The relative frequency distribution of observed values of Y1 and Y2 can be modeled by the probability density function

with time measured in minutes. Find

a P ( Y 1 < 2, Y2 > 1).

b P ( Y 1 ≥ 2Y2).

c P ( Y 1 − Y2 ≥ 1). (Notice that Y1 − Y2 denotes the time spent at the service window.)

Solution:

Step 1 of 2:

Let Y1 be the total time between a customer's arrival at the store and departure from the service window.

And Y2 be the time a customer waits in line before reaching the service window.

The joint density function of Y1 and Y2 is given by,

We have to find

- P(Y1<2, Y2>1).
- P(Y1
- P(Y1