Page 118 - Fundamentals of Communications Systems

3.32 Chapter Three

(c) Show that if X is a Gaussian random variable with mean, m X , and variance,
2
2
2
σ , that φ X (t) = exp( jtm X − t σ /2).
X X
Hint: For any b
1 (x − b)
∞
2
exp − dx = 1. (3.41)
2
2πσ X
−∞ 2 2σ X
(d) Show that if X 1 and X 2 are independent random variables then the char-
(t).
acteristic function of Y = X 1 + X 2 is φ Y (t) = φ X 1 (t)φ X 2
(e) Since the characteristic function is unique, deduce from the results in part
(d) that the sum of two Gaussian random variable is another Gaussian
random variable. Identify m Y and σ Y .
Problem 3.25. The probability of having a cellular phone call dropped in the mid-
dle of a call while driving on the freeway in a “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
Since a call is three time more likely to occur in the daytime than at night, a
person’s time to make a call, T , might be modeled as a random variable with
f T (t) = K 0 ≤ t ≤ 7
= 3K 7 ≤ t ≤ 21
(3.42)
= K 21 ≤ t ≤ 24
= 0 otherwise
where time is measured using a 24-hour clock (i.e., 4:00PM = 16).

(a) Find K.
(b) Find P (A).
(c) What is P (C)?
(d) What is the P (A and C) = P (A∩ C)?