Page 116 - Fundamentals of Communications Systems

3.30 Chapter Three

Problem 3.21. In general, uncorrelatedness is a weaker condition than indepen-
dence and this problem will demonstrate this characteristic.

(a) If X and Y are zero mean independent random variables, prove E[XY ] = 0.
(b) Consider the joint discrete random variable X and Y with a joint PMF of
1
p XY (2, 0) =
3
1
p XY (−1, −1) =
3
1
p XY (−1, 1) =
3
p XY (x, y) = 0 elsewhere (3.35)

What are the marginal PMFs, p X (x), and p Y (y)? Are X and Y independent
random variables? Are X and Y uncorrelated?
(c) Show that two jointly Gaussian random variables which are uncorrelated
(i.e., ρ XY = 0) are also independent. This makes the Gaussian random
variable an exception to the general rule.

Problem 3.22. The probability of having a cellular phone call dropped in the mid-
dle of a call while driving on the freeway in the big city is characterized by
following events

1. A ={call is during rush hour}
2. B ={call is not during rush hour}
3. C ={the call is dropped}
4. D ={the call is normal}
Rush hour is defined to be 7–9:00 AM and 4–6:00 PM. The system is character-
ized with
P (Drop during rush hour) = P (C|A) = 0.3
P (Drop during nonrush hour) = P (C|B) = 0.1.

(a) A person’s time to make a call, T , might be modeled as a random variable
with
1
f T (t) = 0 ≤ t ≤ 24
24
= 0 otherwise (3.36)
where time is measured using a 24-hour clock (i.e., 4:00 PM = 16). Find
P (A).