Chapter 3
Random Variables &
Probability Distributions
Content
1. Two Types of Random Variables
2. Probability Distributions for Discrete
Random Variables
© 2011 Pearson Education, Inc
4.1
Two Types of Random Variables
© 2011 Pearson Education, Inc
Random Variable
A random variable is a variable that assumes
numerical values associated with the random
outcomes of an experiment, where one (and only
one) numerical value is assigned to each sample
point.
© 2011 Pearson Education, Inc
1. Discrete Random Variable(DRV)
Random variables that can assume a countable
number (finite or infinite) of values are called
discrete.
2. Continuous Random Variable
Random variables that can assume values
corresponding to any of the points contained in
one or more intervals (i.e., values that are
infinite and uncountable) are called continuous.
Discrete Random Variable
Examples
Experiment
Random
Variable
Possible
Values
Make 100 Sales Calls
# Sales
0, 1, 2, ..., 100
Inspect 70 Radios
# Defective
0, 1, 2, ..., 70
Answer 33 Questions
# Correct
0, 1, 2, ..., 33
3 identical fair coins
# Number of
HHH, HHT, . . . , TTT
heads
Example 1:
An experiment where 3 identical fair coins are tossed
 Sample space : S= {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
 Define DRV X : Number of heads in a tossSample space : S= {HHH,
HHT, HTH, HTT, THH, THT, TTH, TTT}
 The diagram below shows schematically the numbers that are
assigned by X to each member of the sample space
Graph distribution function p(x)
 P(X = x) or The probability distribution for x.
 Note : the sum of all probabilities is exactly one or ∑ P(X = x) = 1
4.2 Probability Distributions for Discrete Random Variables
The Probability Distribution Function (PDf) for X, p(x) is
a function that assigns each number that X takes, the probability
of equivalent event.
The fourth column,P(X ≤ x), describes the cumulative
distribution for X.
Example 2:
Experiment: Toss 2 coins. Count number of tails.
Answer :
X = Count number of tails.
Probability Distribution
Values, x Probabilities, p(x)
© 1984-1994 T/Maker Co.
0
1/4 = 0.25
1
2/4 = 0.50
2
1/4 = 0.25
Visualizing Discrete
Probability Distributions
Listing
Table
{ (0, 0.25), (1, 0.50), (2, 0.25) }
# Tails
f(x)
Count
p(x)
0
1
2
1
2
1
0.25
0.50
0.25
Graph
p(x)
0.50
0.25
0.00
Formula
x
0
1
2
p (x ) =
n!
px(1 – p)n – x
x!(n – x)!
Example 3:
A)
B)
C)
Properties of the Probability
Distribution Function ( PDf )
1. let P(X= x) = p(x)
p(x) is a probability distribution function or
probability mass function for all x in the range space of X
2. p(x) ≥ 0 for all values of x
3.  p(x) = 1
Note : the individual probabilities are non-negative and add up to one
where the summation of p(x) is over all possible values of x.
Example 4.
Indicate, with a reason, whether the following could represent a discrete probability distribution.
A)
B)
X
X
0
1
3
5
- 0.1
0.1
0.4
0.
-3
2
1
4
0.1
0.2
0.4
0.3
A) P(x) is not a probability distribution function as p(0) = - 0.1 is negative..
B) P(x) is a probability distribution function as the individual probabilities are non negative and add
up to one.
Example 5
Solution:
Example 6.
A box contains 3 white and 2 red balls. 3 balls are drawn without replacement. Define X : the number of red balls
drawn. Determine :
A) The probability distribution for X
B) th ecumulative probability distribution for X.
Solution:
A) Clearly X = 0, 1, 2.
the table below tabulates the probability distribution for X.
X=x
Equivalent event
0
0 red and 3 white balls
1
1 red and 2 white balls
2
2 red and 1 white balls
P( X = x )
Alternative Solution:
A) Clearly X = 0, 1, 2.
Let R : Ball drawn is red (R= 3 balls) W : ball drawn is white (W = 2 balls)
Summary Measures
1. The mean of a discrete random variable X (E(X) or μ):
a weighted average of the possible values that the random
variable can take.
The difference mean of DRV with the sample mean of a group of
observations:
sample mean of a group of observations gives each observation equal
weight, the mean of a DRV weights each outcome xi according to its
E(X) = μ = ∑xP(x)
x
1
2
3
4
5
6
p(x)
0.1
0,2
0,2
0.15
0.15
0.2
Summary Measures
2. Variance is the expected value of the squared variation of a
random
variable from its mean value, in probability
and statistics.
• Informally, variance estimates how far a set of numbers (random)
are spread out from their mean ( average) value and thus from
every other number in the set.
Variance’s symbol: σ2
3.
Standard Deviation (σ) is a measure of how dispersed
the data is in relation to the mean. Low, or small, standard
deviation indicates data are clustered tightly around the mean,
and high, or large, standard deviation indicates data are more
spread out.
2
 
Summary Measures
Calculation Table
x
p(x)
x p(x)
Total
x p(x)
x–
(x – 
(x – p(x)
(x  p(x)
Thinking Challenge
You toss 2 coins. You’re interested
in the number of tails. What are the
expected value, variance, and
standard deviation of this random
variable, number of tails?
Head
Tail
Expected Value & Variance
Solution*
x
p(x)
x p(x)
x–
(x –   (x –  p(x)
0
0.25
0
–1.00
1.00
0.25
1
0.50
0.50
0
0
0
2
0.25
0.50
1.00
1.00
0.25
 = 1.0
 0.50
 0.71
Another solution :
Another solution :
X is the number of Tail
X
0
1
2
1/4
2/4
1/4
Mean of X= E(X) = x p(x)
= ( 0 x ¼) + ( 1 x 2/4) + ( 2 x ¼) = 4/4 = 1
Discrete Uniform Distribution
We may have a situation where the probabilities of each event are
the same.
Example : if we roll a fair die, we assume that the probability of obtaining
each number is …
If X is the random variable ,the probability distribution table is
x
1
2
3
4
5
6
P(X = x)
1
6
1
6
1
6
1
6
1
6
1
6
The probability distribution function (p.d.f.) is
1
P( X  x)  ,
6
x  1, 2, 3, 4, 5, 6
The distribution with equal probabilities is called “uniform”
Discrete Uniform Distribution
Discrete Uniform Distribution
Proof :
Use CAS to find the sum: menu – calculus - sum
Linear change on random variable
E(X) = ∑ (x) p(x) = 3
B) Variance for X is ….
= - 90
C) let Y = 3X , the range for Y = 3,6,9,12.
Linear change on random variable
Practise:
X
1
2
3
4
p(x)
k
4k
9k
16k