Page 94 - Fundamentals of Communications Systems

3.8 Chapter Three

nonzero probabilities since the CDF has jumps. The discrete random variable
has a CDF with the form

N

F X (x) = P (X = a k )U (x − a k ) (3.14)
k=1
where U () is the unit step function, N is the number of jumps in the CDF, and
a k are the locations of the jumps.

3.2.2 Probability Density Function
Definition 3.9 For a continuous random variable X(ω), the PDF is a function f X (x)
defined as
dP (X(ω) ≤ x) dF X (x)
f X (x) = = ∀ x ∈ R (3.15)
dx dx
Since the derivative in Eq. (3.15) can be rearranged to give

lim f X (x) = lim P x − < X(ω) ≤ x + ,
→0 →0 2 2
the PDF can be thought of as the probability “density” in a very small interval
around X = x. Discrete random variables do not have a probability “density”
spread over an interval but do have probability mass concentrated at points. So
the idea of a density function for a discrete random variable is not consistent.
The analogous quantity to a PDF for a continuous RV in the case of a discrete
RV is the probability mass function (PMF)

p X (x) = P (X = x)

The idea of the probability density function can be extended to discrete and
mixed random variables by utilizing the notion of the Dirac delta function
[CM86].

Properties of a PDF
x

■ F X (x) = f X (β)dβ
−∞
■ f X (x) ≥ 0
∞

■ F X (∞) = 1 = f X (β)dβ
−∞

x 2
■ P (x 1 < X ≤ x 2 ) = f X (β)dβ
x 1
Again, if there is no ambiguity in the expression, the PDF of the random
variable X is written as f (x) and if there is no ambiguity about whether a
RV is continuous or discrete, the PDF is written p(x). Knowing the PDF (or
equivalently the CDF) allows you to completely describe any random event
associated with a random variable.