3. DISCRETE RANDOM
VARIABLE
3.1 Expected Value of Discrete
Random Variables
3.2 The Binomial Random Variable
3.1 Expected Value of Discrete Random
Variables
• The mean, or expected value, of a discrete random
variable x is
  E ( x)   xp( x)
2
• The population variance  is defined as a average of
the squared distance of x from the population mean  .
• The variance of random variable x is
 2  E[( x   2 )]  ( x   ) p( x)
• The standard deviation of a discrete random variable is
equal to the square root of the variance, i.e., to
  2
Figure 3a.
Shape of two probability distribution for a
discrete random variable x
3.2 The Binomial Random Variable
3.2.1 Characteristics of a Binomial Random Variable
•
•
•
•
•
The experiment consist of n identical trials.
There are only two possible outcomes on each trial. We will
denote one outcomes by S (for Success) and the other by F
(for Failure).
The probability of S remains the same from trial to trial. This
probability is denoted by p, and the probability of F is denoted
by q. Note that q=1- p.
The trials independent.
The binomial random variable x is the number of S’s in n
trials.
3.2.2 The Binomial Probability Distribution
 n x n  x
p( x)    p q
 x
(x = 0, 1, 2, …, n)
Where:
p = Probability of a success on a single trial
q = 1- p
n = Number of trials
n
n!
  
 x  x!(n  x)!
The Binomial probability distribution is so named because the
probabilities, p(x), x = 0, 1, …,n, are terms of the binomial expansion,
(q+p)n.
3.2.3 Mean, Variance and Standard Deviation for a
Binomial Random Variable
Mean :   np
Variance :  2  npq
Standard :   npq
Figure 3b.
Graph of Binomial Probability
distribution or n = 20 and p = 0.6
Figure 3c.
The Binomial Probability Distribution
for n = 20 and p = 0.6