Page 107 - Fundamentals of Communications Systems

Review of Probability and Random Variables 3.21

where |J | is the Jacobian defined as

∂x 1 ∂x 2
... ∂x n
∂y 1 ∂y 1 ∂y 1

∂x 1 ∂x 2
... ∂x n
∂y 2 ∂y 2 ∂y 2
|J |= .

. . . . .

∂x 1 ∂x 2 ... ∂x n
∂y n ∂y n ∂y n
EXAMPLE 3.17
A one-to-one transformation which is very important in communications is

2
Y 1 = X + X 2 X 1 = Y 1 sin(Y 2 )
1 2
Y 2 = tan −1 X 1 X 2 = Y 1 cos(Y 2 )
X 2
The Jacobian of the transformation is given by |J |= Y 1 which results in the joint
density of the transformed random variables being expressed as
(y 1 sin(y 2 ), y 1 cos(y 2 ))
p Y 1 Y 2 (y 1 , y 2 ) = y 1 p X 1 X 2

3.3.5 Central Limit Theorem
Theorem 3.6 If X k is a sequence of independent, identically distributed (i.i.d.) random
2
variables with mean m x and variance σ and Y n is a random variable defined as
n
1
Y n = √ (X k − m x )
nσ 2
k=1
then
2
1 y
(y) = √ exp − = N (0, 1)
lim p Y n
n→∞ 2π 2
Proof: The proof uses the characteristic function, a power series expansion, and
the uniqueness property of the characteristic function. Details are available in
most introductory texts on probability [DR87, LG89, Hel91].

The central limit theorem (CLT) implies that the sum of arbitrarily dis-
tributed random variables tends to a Gaussian random variable as the number
of terms in the sum gets large. Because many physical phenomena in communi-
cations systems are due to interactions of large numbers of events (e.g., random
electron motion in conductors or large numbers of scatters in wireless propa-
gation), this theorem is one of the major reasons why the Gaussian random
variable is so prevalent in communication system analysis.