Page 112 - Fundamentals of Communications Systems

3.26 Chapter Three

Radar designers on the other hand like to quote performance in the following
two parameters (because they can easily be measured)

■ P (DT |TA) = probability of detecting a target when a target is absent.
■ P (DT |TP) = probability of detecting a target when a target is present.
Imagine that you are a high-priced consultant for the Federal Aviation Ad-
ministration (FAA) and that the FAA has the following requirements for its
next generation ATCR

■ P (TA|DT ) = 0.01
■ P (TP|ND) = 0.0001

A detailed study shows that planes are present only 1% of the time, P (TP) =
0.01. A contractor, Huge Aircrash Co., has offered a radar system to the gov-
ernment with the following specifications
■ P (DT |TA) = 0.00005
■ P (DT |TP) = 0.9

Would you recommend the government purchase this system and why?

Problem 3.9. The following are some random events
1. The sum of the roll of two dice.
2. The hexadecimal representation of an arbitrary 4 bits in an electronic
memory.
3. The top card in a randomly shuffled deck.
4. The voltage at the output of a diode in a radio receiver.
Determine which events are well modeled with a random variable. For each
random variable determine if the random variable is continuous, discrete, or
mixed. Characterized the sample space and the mapping from the sample space
if possible.
Problem 3.10. A random variable has a density function given as

f X (x) = 0 x < 0
= K 1 0 ≤ x < 1
= K 2 1 ≤ x < 2
= 0 x ≥ 2

(a) If the mean is 1.2, find K 1 and K 2 .
(b) Find the variance using the value of K 1 and K 2 computed in (a).
(c) Find the probability that X ≤ 1.5 using the value of K 1 and K 2 computed
in (a).