Module 3: Random Variables
Lecture – 2: Probability Distribution of Random Variables
Probability Distribution of a Random variable
Probability distribution of a Random Variable (RV) is a function that provides a complete
description of all possible values that the RV can take along with their probabilities over the
range of minimum and maximum possible values (in a statistical sense) of that RV.
Probability Distribution of a discrete random variable
Probability distribution of a discrete random variable specifies the probability of each possible
value of the random variable. Being probabilities, the distribution functions of discrete random
variables are concentrated as a mass for a particular value, and generally known as Probability
Mass Function (PMF). In figure 1, the PMF of a discrete random variable X is plotted against
different possible values i.e, x. Thus, the Probability Mass Function (PMF) is the probability
distribution of a discrete random variable and generally denoted by px(x). It indicates the
probability of the value X = x taken by X.
Properties of PMF
The properties of PMF are:

p X x   0

 p x   1
for all X
X
It may be noted that if in a particular case, it is certain that the outcome is only c, then
In
the
case
of
mutually
exclusive
outcomes,
p X  c   P  X  c  1 .


 
 
x1 , x2 ,......xn , p x1  x2  ...  xn  p X x1  p X x2  ...  p X  xn  .
Example problem of PMF
Prob. The number of floods recorded per year at a gauging station in Italy are given in the Table
1. Find the PMF and plot it. (Kottegoda and Rosso, 2008)
1 Table 1: Number of flood occurrences per year from 1939 to 1972
Number of floods in a
year
Number of occurrences
Number of floods
in a year
Number of
occurrences
0
0
5
4
1
2
6
1
2
6
7
4
3
7
8
1
4
9
9
0
Soln. Total number of floods is 34.
Let X denote the number of occurrence of flood. The probabilities for different number of floods
can be obtained as follows.
2
6
7
9
4
p X  X  0   0; p X  X  1  ; p X  X  2   ; p X  X  3  ; p X  X  4  
; p X  X  5  ;
34
34
34
34
34
p X  X  6 
4 
1

; pX  X 
  0; p X  X  8   ;
34
34 
34

1
and p X  x   0 for al x  9 The obtained PMF are plotted in the Fig. 1.
Fig. 1. Probability Mass Function (PMF) of flood occurrences X
per year at the gauging station
2 Some standard Probability Mass Functions (PMFs)
Some standard PMFs are:

Binomial Distribution

Multinomial Distribution

Negative Binomial Distribution

Geometric Distribution

Hypergeometric Distribution

Poisson Distribution
These distributions will be discussed in subsequent lectures.
Probability Distribution of a continuous random variable
Probability distribution of a continuous random variable specifies continuous distribution of
probability over the entire feasible range of the random variable. In contrast to the discrete
random variable, the probability of a continuous random variable is distributed over the entire
range of the RV and at a particular value the magnitude of the distribution function can be
treated as density. In physics, density functions can be integrated to obtain mass; similarly
probability density functions can be integrated to obtain probabilities. Hence, the distribution
function of continuous random variables is generally known as probability density function
(pdf).
Thus, A Probability Density Function (pdf) is the probability distribution of a continuous random
variable. It is conventionally denoted by f X  x  . In figure 2, the pdf of a continuous random
variable X is plotted against x.
Fig. 2 Probability Density Function (pdf) of a continuous random variable X
3 Conditions for valid pdf
The conditions for a valid pdf are
1. The pdf is a continuous nonnegative function for all possible values of x.
f X  x  0
for all X
2. The total area bounded by the curve and the x axis is equal to 1. In figure 3, the shaded
area under the curve is equal to 1.

 f  x  dx  1
 X
Fig. 3 Probability Density Function (pdf) of X
It may be noted that when a pdf is graphically portrayed, the area under the curve between two
limits, x1 and x2 (such that x2 > x1 ) gives the probability that the random variable X lies in the
interval x1 to x2. In figure 4, the shaded portion gives the probability of the random variable X
lying in the interval x1 to x2 .
x2
P  x1  X  x2    f X  x  dx  1
x1
4 1
Fig. 4 Probability Density Function (pdf) of X
Example problem on pdf
Prob. (Ang and Tang,1975) A random variable X has a pdf of the form
f X  x   x 2
0  x  10
0
elsewhere
(a) Under what condition is this function a valid pdf ?
(b) What is the probability of X being greater than 5?
Solution:
(a) To satisfy all properties of pdf,
10
 x dx  1
2
0
10
 x3 
or ,     1
 3 0
3
or ,  
1000
(b) Now the probability
P  X  5  1  P  X  5  1 
5

0
3x 2
53
dx  1 
 0.875
1000
1000
5 Some standard probability distribution functions (pdfs)
Normal or Gaussian Distribution
Normal Distribution is a continuous probability distribution function with parameters μ (mean)
and σ2 (variance) and its probability density function can be expressed as:
f X x  
1
2
2
e

 x   2
2 2
for    x  
Figure 5 shows 3 normal distribution functions with different parameters.
f X x 
x
Fig. 5 Normal distribution functions
Exponential Distribution
Exponential Distribution is the probability distribution function with parameter λ, and its
probability density function is expressed as: f X  x   e  x
for x  0
0
otherwise
Figure 6 shows 3 exponential distribution functions with different parameters.
6 Fig. 6 Exponential distribution functions
Gamma Distribution
Gamma distribution is the probability distribution function with parameters α and β (where α>0,
β>0) and its probability density function is expressed as,
f X x  
x 1
  
0
e x / 
for x  0
otherwise

where     x 1e  x dx
0
When α is an positive integer,      1!
Figure 7 shows 3 gamma distribution functions with different parameters.
7 Fig. 7 Gamma distribution functions
8