Class notes SST 204: PROBABILITY AND STATISTICS 1

KENYATTA UNIVERSITY


DIGITAL SCHOOL OF VIRTUAL AND OPEN LEARNING
IN COLLABORATION WITH
SCHOOL OF PURE & APPLIED SCIENCES
DEPARTMENT: MATHEMATICS AND ACTUARIAL SCIENCE




SST 204: PROBABILITY AND STATISTICS I




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, LESSON ONE
RANDOM VARIABLES
1.1 Introduction
In this lesson we will discuss the definition of a random variable, types of
random variables and their probability distributions.
1.2 Lesson Learning Outcomes
By the end of this lesson the learner will be able to:
i. Define a random variable
ii. State types of random variables
iii. Obtain the probability distributions of discrete and continuous
random variables.
1.3 Random variables
Let S be a sample space representing the outcomes of a statistical experiment.
Then we can define a random variable as follows:
A random variable X is a real valued function defined on S, that is
X : S→R

Capital letters are used to denote random variables while small letters are used to
denote respective values of the random variables.
Example 1.1: Suppose that three boys are selected at random from a school
parade and each is asked whether he smokes (S) or he does not (N). Then the
sample space of this random experiment is given by
S= { SSS , SSN , SNS , NSS , SNN , NSN , NNS , NNN }

Let X denote the number of smokers among three chosen boys. Then
X ( SSS )=3 , X ¿

X ( SNN )=X ( NSN )=X ( NNS )=1 , X ( NNN )=0 .

Therefore, X is a random variable which takes the values 0,1,2,3.




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,Example 1. 2 : Suppose that a real number is selected at random in the closed
interval [ 0,2 ] . Let X denote the number so chosen. Then X is a random variable
with possible values x , 0 ≤ x ≤ 2.
There are two types of random variables namely discrete and continuous random
variables.
1.3.1 : Discrete Random Variables


A random variable is said to be discrete if it assumes only a finite or countable
number of values on the real line. e.g , the random variable described in
example 1.1
A discrete random variable assumes each of its values with a certain
probability. In example 1 if we assume that all the outcomes are equally likely
then the random variable X takes the values 0,1,2,3 with the following
probabilities:

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P(X=0) =P{NNN} = 8 ,
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P(X=1) =P{SNN, NSN,NNS} = 8 ,
3
P(X=2) =P{SSN, SNS,NSS} = 8 ,
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and P(X=3) =P{SSS} = 8 .
We can write these probabilities in a table form as follows:


x 0 1 2 3
P(X=x) 1 3 3 1
8 8 8 8

NB: P(X=0) + P(X=1) + P{X=2) +P (X=3) =1.
The above table represents a probability distribution of the random variable X.
Let X be a discrete random variable, then the probability distribution of X is a
real valued function f(x) of x, defined by


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, f ( x )=P ( X =x )
and satisfying the following conditions:
( i ) f ( x ) ≥ 0 for all x

x=∞
( ii ) ∑ f ( x )=1.
x=−∞



Example 1.3: A digit is selected at random from among the digits
0,1,2,3,4,5,6,7. Let X denote the digit so selected. What is the probability
distribution of X?




Solution
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P ( X=x ) = , x=0,1,2,3,4,5,6,7
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and P ( X=x ) =0 , x ∉{0,1,2,3,4,5,6,7}
Therefore, the probability distribution of X is by




{
1
,∧x=0,1,2,3,4,5,6,7
f ( x )= 8
0 ,∧otherwise
¿9
Obviously f ( x ) ≥ 0 and ∑ f ( x )=1.
x=0


A probability distribution given by


{
1
,∧x=1,2 , .. . , n
f ( x )= n
0 ,∧otherwise

is known as a discrete uniform distribution.
Example 1. 4 Let X be a discrete random variable whose set of values is the set of
all non-negative integers. Show that the function f ( x ) given by



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