Question

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 Y Y 1 2 ≥ . The

relative frequency distribution of observed values of Y1 and Y2 can be modeled by the joint

probability density function as

( ) 1

2 1

1 2

0 , , 0 elsewhere.

y

f e y y

y y

−  ≤ ≤ ≤∞ = 



Then, the random variable interest is the time spent at the service window given by

UYY = −1 2 .

a) Find the probability density function of U .

[14 Marks]

b) Find E U( ) .

[3 Marks]

c) Find V U( ).

[3 Marks]