Discrete Random Variables 
Probability Distributions
A probability distribution is similar to
a frequency distribution or a histogram. Defined characteristics
of a population selected randomly is called a random variable and
when the values of this variable is measurable we can determine its mean
or average or expected value and also its variance and standard
deviation.
Review: frequency
distribution, mean and variance
1. Know what is meant by a random
variable.
A random variable is a numerical quantity whose value is determined
by chance.
In research one is often asked to study a population, the researchers
must therefore define or select characteristics of the populations that
they which to study or measure, the characteristics of a population that
one wishes to study is called a random variable and its possible
values is the sample space.
The values of the sample space is subject to chance and is therefore
determined randomly, these values are said to have been occurred or observed.
Examples of Random Variables:
Your income next year 
Your score on the next exam 
Tomorrow's high temperature 
The number of errors in a phone book 
How long you wait for a soda 
IBM's quarterly dividend 
Length of a movie line 
The height of the next person you see 
Amount of money you will lose 
Your GPA for the present term 
2. Know what is meant by a continuous or discrete
random variable.
A discrete random variable is one that can assume only integer
(whole number, 0, 1, 2, 3, 4, 5, 6, etc. ) Values.
Example: Response with Yes  No values, Maximum Number
of adult that can fit into a car (4 , 5, 6 or 7), Year (2002), etc.
A continuous random variable is one that can assume any value
over a continuous range of possibilities.
Example: Time of day (12:31:24 p.m.), Temperature (60.3145^{0}
F, Weight (154.25 pounds), etc.
A continuous random variable may be reported along an interval
which show the range of possible values, sample space, such as the for
the random continuous variable, x, the height of a grown man: on estimate
would be 4 feet < x < 7 feet (Interval).
3. Know how to determine the probability
distributions of discrete random variables.
The probability distribution of a discrete random variable is similar
to a frequency distribution of that variable. This distribution may be
illustrated or represented by either a table or a graphical presentation
such as a histogram.
Example: Consider an experiment
to count the number of customers arriving during a specific time interval
(say, number arriving at 10 minutes intervals). The discrete random variable
would be the number of arrivals during the time interval, let's say that
the possible numbers arriving is either, 0, 1, 2, 3, 4, 5, 6, and 7 or
greater. A results of such an experiment would look something like this:
The Pr[x] or P(x) or frequency of x is the cell frequency divided by
total number of observation.
Frequency Distribution Table is shown in Table 5.1
Table 5.1 Number of Arrivals Probability Distribution Table
Number
of Arrival, x
(1) 
Numbers
of 10 minutes Interval with x number of arrivals, f
(2) 
Relative
Frequency or Probability of x, f/N
(3)=(2)/N 
0 
18 
0.3529 
1 
18 
0.3529 
2 
8 
0.1569 
3 
3 
0.0588 
4 
2 
0.0392 
5 
1 
0.0196 
6 
0 
0 
7 or more 
1 
0.0196 

N=51 
1 
Figure 5.1 Probability Distribution Plot.
Workshop Problem: Probability Distribution.
Find the probability distribution of x, if x represents a code for the
salary range of employee, assume x = 5 is the lowest range and x = 30 is
the highest range.
Salary
Range, x 
Number
of Salaries, f 
P(x) =
f/N 
1 
12 

2 
10 

3 
7 

4 
3 

5 
4 

6 
3 


Total = N= 
1 
Mean and Variance of Random Variables
The mean or expected value of a
random variable is the sum of each values of the variable times its corresponding
probability, p(x).
The expected value of a random variable
is
considered its mean.
1. Know how to determine the mean
or expected value of a discrete random variable.
The computation used to calculate the mean or expected value of a random
variable is similar to that used to find the mean of a grouped data.
The expected value of a discrete random variable, X, denoted by ,
is the weighted average of that variable's possible values, where the respective
probabilities are used as weights.
The expected value is also denoted by E(x).
Example: Using example above to compute the Expected Value of
x.
From worksheet below, the expected value is 1.2353.
Number
of Arrival, x
(1) 
Numbers
of 10 minutes Interval, f
(2) 
Relative
Frequency or Probability of x, Pr(x)=f/N
(3)=(2)/N 
x Pr(x) 
0 
18 
0.3529 
0 
1 
18 
0.3529 
0.3529 
2 
8 
0.1569 
0.3137 
3 
3 
0.0588 
0.1765 
4 
2 
0.0392 
0.1569 
5 
1 
0.0196 
0.098 
6 
0 
0 
0 
7 or more 
1 
0.0196 
0.1374 

N=51 
1 
=
1.2353

2. Know how to compute the variance and standard
deviation from a frequency distribution.
The variance of a discrete random variable is determined by the following
formulas, (2) Is preferred for computational ease:
(1) Variance = ,
where P(x) is the probability or relative frequency of x.
(2) Variance =
Example: calculate the variance
of the random variable above:
Number
of Arrival, x
(1) 
Numbers
of 10 minutes Interval, f
(2) 
Relative
Frequency or Probability of x, Pr(x)=f/N
(3)=(2)/N 
x Pr(x) 
x^{2}
Pr(x) 
0 
18 
0.3529 
0 
0 
1 
18 
0.3529 
0.3529 
0.3529 
2 
8 
0.1569 
0.3137 
0.6275 
3 
3 
0.0588 
0.1765 
0.5294 
4 
2 
0.0392 
0.1569 
0.6275 
5 
1 
0.0196 
0.098 
0.4902 
6 
0 
0 
0 
0 
7 or more 
1 
0.0196 
0.1374 
0.9608 

N=51 
1 
=
1.2353

=
=
3.5882 
Variance =
= 3.58821.5260=2.0622
And standard deviation =
Worksheet: Use the worksheet or functions below to show the probability
distribution, mean, variance and standard deviation of the random variable
x below:
x 
0 
1 
2 
3 
4 
5 
6 
Frequency 
23 
43 
50 
56 
45 
34 
24 
Worksheet for Computing the Probability
Distribution, mean and variance of a Discrete Random Variable, x.
Number
of Arrival, x
(1) 
Numbers
of 10 minutes Interval, f
(2) 
Relative
Frequency or Probability of x, Pr(x)=f/N
(3)=(2)/N 
x Pr(x)
(4) 
x^{2}
Pr(x)
(5) =
(1) x (4) 









































N= 
1 
= 
(7)= 


(6) 
Variance =
(8) =(5)  (6)=
Standard Deviation = = 
