Page 100 - Fundamentals of Communications Systems

3.14 Chapter Three

3.3 Multiple Random Variables
In communication engineering, performance is often determined by more than
one random variable. To analytically characterize the performance, a joint
description of the random variables is necessary. All of the descriptions of a
single random variable (PDF, CDF, moments, etc.) can be extended to the sets
of random variables. This section highlights these extensions.

3.3.1 Joint Density and Distribution Functions
Again the random variables are completely characterized by either the joint
CDF or the joint PDF. Note, similar simplifications in notation will be used
with joint CDFs and PDFs, as was introduced in Section 3.2, when no ambiguity
existed.

Definition 3.10 The joint CDF of two random variables is
F XY (x, y) = P ({X ≤ x}∩{Y ≤ y}) = P (X ≤ x, Y ≤ y)

Properties of the Joint CDF
■ F XY (x, y) is a monotonically nondecreasing function, i.e.,
and F XY (x 1 , y 1 ) ≤ F XY (x 2 , y 2 )
x 1 < x 2 y 1 < y 2 ⇒
■ 0 ≤ F XY (x, y) ≤ 1

■ F XY (−∞, −∞) = P (X ≤−∞, Y ≤−∞) = 0 and F XY (∞, ∞) = P (X ≤
∞, Y ≤∞) = 1
■ F XY (x, −∞) = 0 and F XY (−∞, y) = 0
■ F X (x) = F XY (x, ∞) and F Y (y) = F XY (∞, y)
■ P (x 1 < X ≤ x 2 , y 1 < Y ≤ y 2 ) = F XY (x 2 , y 2 ) − F XY (x 1 , y 2 ) − F XY (x 2 , y 1 ) +
F XY (x 1 , y 1 )

Definition 3.11 For two continuous random variables X and Y , the joint PDF,
f XY (x, y), is
2
2
∂ P (X ≤ x, Y ≤ y) ∂ F XY (x, y)
f XY (x, y) = = ∀ x, y ∈ R
∂x∂y ∂x∂y
Note, jointly distributed discrete random variables have a joint PMF,
p XY (x, y), and as the joint PMF and PDF have similar characteristics this
text often uses p XY (x, y) for both functions.

Properties of the Joint PDF or PMF
y x

■ F XY (x, y) = p XY (α, β)dαdβ
−∞ −∞
■ p XY (x, y) ≥ 0

∞ ∞
■ p X (x) = p XY (x, y)dy and p Y (y) = p XY (x, y)dx
−∞ −∞
x 2 y 2

■ P (x 1 < X ≤ x 2 , y 1 < Y ≤ y 2 ) = p XY (α, β)dαdβ
x 1 y 1