Page 115 - Fundamentals of Communications Systems

Review of Probability and Random Variables 3.29

(d) How many independent locations would you have to try to ensure a 90%
chance of getting your cellular phone to work?

Problem 3.18. The function randn in Matlab produces realizations of a zero
mean–unit variance Gaussian random variable each time it is called. This
problem leads you to ways to use this function in more general ways. Assume
X is a Gaussian random variable with zero mean and unit variance.
(a) Define a random variable Y = X + b, find f Y (y).
(b) Define a random variable Y = aX, where a > 0, find f Y (y) (use
Example 3.11).
(c) How can you transform X, Y = g(X), so that Y is a Gaussian random
variable with m Y = b and σ Y = a?
(d) Test out your answer by invoking randn 1000 times and transforming these
1000 samples as you propose in c) with m y = 6 and σ y = 2. Plot a histogram
of the transformed output.

Problem 3.19. This problem gives both a nice insight into the idea of a corre-
lation coefficient and shows how to generate correlated random variables in
simulation. X and W are two zero mean independent random variables where
2
2
2
E[X ] = 1 and E[W ] = σ . A third random variable is defined as Y = ρX +W,
W
where ρ is a deterministic constant such that −1 ≤ ρ ≤ 1.
(a) Prove E[XW] = 0. In other words prove that independence implies uncor-
relatedness.
2
(b) Choose σ 2 such that σ = 1.
W Y
2
(c) Find ρ XY when σ W is chosen as in part (b).
Problem 3.20. You are designing a phase-locked loop as an FM demodulator. The
requirement for your design is that the loop bandwidth must be greater than
5 kHz and less than 7 kHz. You have computed the loop bandwidth and it is
given as

3
B L = 4R + 2000
where R is a resistor in the circuit. It is obvious that choosing R = 10 will solve
your problem. Unfortunately, resistors are random valued.

(a) If the resistors used in manufacturing your FM demodulator are uniformly
distributed between 9 and 11 , what is the probability that the design will
meet the requirements?
(b) If the resistors used in manufacturing your FM demodulator are Gaussian
random variables with a mean of 10 and a standard deviation of 0.5 ,
what is the probability that the design will meet the requirements?