Page 93 - Fundamentals of Communications Systems

Review of Probability and Random Variables 3.7

F (x) F (x)
x
x
1 • 1
• 0
• 0
• 0
0
x x
(a) (b)

F (x)
1 x •
0
•
• 0
0

x
(c)
Figure 3.2 Example CDF for (a) discrete RV, (b) continuous RV, and (c) mixed RV.

Again to simplify the notation when no ambiguity exists, the CDF of the
random variable X will be written as F X (x) = P (X ≤ x). Figure 3.2 shows
example plots of the CDF for discrete, continuous and mixed random variables.
The discrete random variable has a CDF with a stairstep form, the steps occur
at the points of the possible values of the random variable. The continuous
random variable has a CDF, that is a continuous function and the mixed random
variable has a CDF containing both intervals where the function is continuous
with non-zero derivative and points where the function makes a jumps.

Properties of a CDF
2
■ F X (x) is a monotonically increasing function , i.e.,
x 1 < x 2 ⇒ F X (x 1 ) ≤ F X (x 2 )

■ 0 ≤ F X (x) ≤ 1
■ F X (−∞) = P (X ≤−∞) = 0 and F X (∞) = P (X ≤∞) = 1
■ A CDF is right continuous, i.e., lim h→0 F X (x +|h|) = F X (x)
■ P (X > x) = 1 − F X (x) and P (x 1 < X ≤ x 2 ) = F X (x 2 ) − F X (x 1 )
■ The probability of the random variable taking a particular value, a, is given as

P (X = a) = F X (a) − lim F X (x −|h|)
h→0
Continuous random variables take any value with zero probability since the
CDF is continuous, while discrete and mixed random variables take values with

2 This is sometimes termed a nondecreasing function.