Probability distribution of a random variable module

11
MATHEMATICS
(STATISTICS AND PROBABILITY)
Probability
Distribution of a
Random Variable
DAISY RODRIGUEZ ACAS
ILIGAN CITY NATIONAL HIGH SHOOL (SENIOR HIGH SHOOL
DEPARTMENT)

MODULE OVERVIEW
This module on Statistics and Probability is an instructional material
designed to facilitate and enhance the teaching-learning process in Inferential
Statistics intended for Senior High School students. Furthermore, it aims to
supplement and strengthen knowledge and skills both required and acquired in
this course.
This module consists the basic concepts/theories/principles and
applications of Inferential Statistics. At the end of each lesson are exercises
that include practical applications of the statistical principles.
Statistics and Probability was prepared primarily to address the
perennial need in the one-semester course in Inferential Statistics that is the
inadequacy of instructional materials in statistics.
This module was prepared with the hope that concepts in inferential
statistics will be easy to comprehend. Every basic concept and method is
therefore fully explained in language that is easy to understand. To emphasize
applications, the explanations are supplemented with various examples and
exercises.

LEARNING COMPETENCIES
CONTENT STANDARDS:
In this module, the learner is able to demonstrate understanding of key
concepts of random variables and probability distributions.
PERFORMANCE STANDARDS:
The learner is able to apply an appropriate random variable for a given
real-life problem (such as in decision making and games of chance).
LESSON AND COVERAGE:
LESSON LEARNING COMPETENCIES
1
Random Variable
 The learner is able to illustrate a random
variable (discrete and continuous).
 The learner is able to distinguish between
a discrete and a continuous random
variable.
 The learner is able to find the possible
values of a random variable.
2
Discrete Probability
Distribution
 The learner is able to illustrate a
probability distribution for a discrete
random variable and its properties.
 The learner is able to construct the
probability mass function of a discrete
random variable and its corresponding
histogram.
 The learner is able to compute
probabilities corresponding to a given
random variable.

Multiple Choice: Choose the letter that you think the best
answer to each of the following questions. Write the letter of your
choice on the space provided.
______ 1. Which of the following is a discrete random variable?
a. the average amount of electricity consumed
b. the number of persons eating seafood in a restaurant
c. the percentage of the students who passed the exam
d. the average weight of female athletes
______ 2. If a pair of dice is rolled, which is not a possible value of the random
variable for the sum of the points on the upturned faces of the dice?
a. 1 b. 4 c. 7 d. 11
______ 3. A pair of dice is rolled. Let Y be the random variable denoting the
sum of the points on the upturned faces of the dice. Find the values of the
random variable Y.
a. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 c. 1, 3, 5, 7, 9 and 11
b. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 d. 2, 4, 6, 8, 10 and 12
______ 4. The following are continuous random variable except _______.
a. weight of persons c. income/money
b. height of a person d. number of heads in 3 tosses of a fair coin
______ 5. A function whose value is a real number determined by each element
in the sample space – the set of all possible outcomes of an experiment.
a. random variable c. density function
b. probability distribution d. histogram
______ 6. The sample space of rolling a die and tossing a coin simultaneously.
a.  
T
H
T
H
T
H
S 6
,
5
,
4
,
3
,
2
,
1

b.  
T
H
T
H
T
H
T
H
T
H
T
H
S 6
,
6
,
5
,
5
,
4
,
4
,
3
,
3
,
2
,
2
,
1
,
1
 answer
c.  
T
T
T
H
H
H
S 6
,
5
,
4
,
3
,
2
,
1

d.  
,
6
,
5
,
4
,
3
,
2
,
1 H
H
H
T
T
T
S 
______ 7. Which formula gives the probability distribution shown by the table?
X 1 3 5
P(X) 6
1
3
1
2
1
a.  
12
1


X
X
P b.  
6
X
X
P  c.  
18
2


X
X
P d.  
6
1

X
P
______ 8. If  
6
X
X
P  , what are the possible values of X for it to be a
probability distribution?
a. 1, 2, 3 b. 0, 2, 3 c. 2, 3, 4 d. 1, 1, 2
______ 9. What makes the distribution below a probability distribution?
a. The random variable is equally distributed.
b. The sum of all values of P(x) is equal to one.
c. The values of a probability distribution are probabilities.
d. The values of the P(x) are numbers on the interval from 0 to 1.
X 0 2 4 6 8
P(X) 6
1
6
1
3
1
6
1
6
1
PRE-TEST.

______ 10. Four coins are tossed. Let Y be the random variable representing
the number of heads that occur. The probability distribution of the random
variable Y is _______.
a.
b.
c.
d.
𝑦 = 𝑜𝑓 ℎ𝑒𝑎𝑑𝑠 0 1 2 3 4
P(x) 4/16 6/16 1/16 4/16 1/16
𝑦 = 𝑜𝑓 ℎ𝑒𝑎𝑑𝑠 0 1 2 3 4
P(x) 1/16 4/16 6/16 4/16 1/16
𝑦 = 𝑜𝑓 ℎ𝑒𝑎𝑑𝑠 0 1 2 3 4
P(x) 6/16 4/16 1/16 4/16 1/16
𝑦 = 𝑜𝑓 ℎ𝑒𝑎𝑑𝑠 0 1 2 3 4
P(x) 1/16 4/16 1/16 4/16 6/16

PRE-TEST
1. B
2. A
3. A
4. D
5. A
6. B
7. A
8. A
9. B
10. B
ANSWER KEY.

PROBABILITY DISTRIBUTION OF A RANDOM VARIABLE
LESSON 1: RANDOM VARIABLE
CODE: M11/12SP-IIIa-1, M11/12SP-IIIa-2 and M11/12SP-IIIa-3
OBJECTIVES:
In this lesson, the students will be able to define and illustrate a random
variable, find the possible values of a random variable and distinguish
between a discrete and a continuous random variable.
DISCUSSION:
Probability distributions play an important role in the application of
Statistics. They are used to model the behavior of many variables of interest.
What do you know about me?
Directions: Working in groups, determine the sample space illustrated in
each situation below.
a. Among the applicants of an immersion, there are only two vacant
positions. Three males and two female applicants qualify for the final
interview. Because of scheduling problems, only two applicants can be
interviewed immediately. The personnel manager selects these two by
assigning a numbered ticket to each applicant, and then randomly
drawing two tickets.
b. An experiment consists of asking 2 parents if they are in favor of two
more years in the K-12 curriculum. Using “N” for no, “Y” for yes and “U”
for undecided.
c. A farmer has three fields, each of which must be planted with rice or
corn. He does not plant the same crop in all three fields. Generating a
tree diagram, list all the elements of the sample space S.
Activity 1.

Definition (Random Variable)
A random variable is a function whose value is a real number
determined by each element in the sample space – the set of all possible
outcomes of an experiment and is denoted by S.
The variables that are measured in most scientific studies, whose values
occur by chance, are called to be “random variables”. These are assumed to
follow a probability distribution when used in a statistical analysis.
Do you still remember me?
Directions: List the sample space of the following experiments.
Experiment Sample Space
1. Tossing three coins.
2. Rolling a die and tossing a coin
simultaneously.
3. Drawing a spade from a deck of cards.
4. Getting a defective item when two items
are randomly selected from a box of two
defective and three non-defective items.
5. Drawing a card greater than 7 from a deck
of cards.
Remark:
Capital letters like X , Y , or Z are used to denote a random variable.
Examples:
1. A coin is tossed 3 times. Let X be the random variable denoting the number
of heads. Find the values of the random variable X .
Solution: To do this, first identify all the possible outcomes of the given
statistical experiment. Determine the specific random variable defined in
the problem i.e.
Outcomes TTT TTH THT THH HTT HTH HHT HHH
𝑥 = 𝑛𝑜. 𝑜𝑓 ℎ𝑒𝑎𝑑𝑠 0 1 1 2 1 2 2 3
Therefore, the possible values of the random variable X are 0, 1, 2 & 3.
Activity 2.

2. A pair of dice is rolled. Let Y be the random variable denoting the sum of
the points on the upturned faces of the dice. Find the values of the random
variableY .
Solution: The specific random variable and the possible outcomes of the
given statistical experiment are as follows:
Outcome
s
(1,1
)
(1,2
)
(1,3
)
(1,4
)
(1,5
)
(1,6
)
(2,6
)
(3,6
)
(4,6
)
(5,6
)
(6,6
)
(2,1
)
(2,2
)
(2,3
)
(2,4
)
(2,5
)
(3,5
)
(4,5
)
(5,5
)
(6,5
)
(3,1
)
(3,2
)
(3,3
)
(3,4
)
(4,4
)
(5,4
)
(6,4
)
(4,1
)
(4,2
)
(4,3
)
(5,3
)
(6,3
)
(5,1
)
(5,2
)
(6,2
)
(6,1
)
sum
Y  2 3 4 5 6 7 8 9 10 11 12
Therefore, the possible values of the random variable Y are 2, 3, 4, 5,
6, 7, 8, 9, 10, 11 and 12.
How do you solve me?
Directions: Do what is indicated.
1. Suppose three cell phones are tested at random where D represent the
defective cell phone and N represent the non-defective cell phone. Let
X represents the number of defective cell phones that occur. , show the
values of the random variable X .
2. Two balls are drawn in succession without replacement from an urn
containing 5 red balls and 6 blue balls. Let Z be the random variable
representing the number of blue balls. Find the values of the random
variable Z .
3. Four coins are tossed. Let Y be the random variable representing the
number of heads that occur. Find the values of the random variableY .
Activity 3.

4. A shipment of five computers contains that are slightly defective. If a
retailer receives three of these computers at random, list the elements
of the sample space S using the letters D and N for defective and non-
defective computers, respectively. To each sample point, assign a value
x of the random variable X representing the number of computers
purchased by the retailer which are slightly defective.
Definition (Discrete/Continuous Random Variable)
If a random variable takes only a finite number of values or if the set of
possible outcomes is countable, then it is called discrete random variable.
Otherwise, if it takes on any value in a continuous scale then it is called
continuous random variable.
Some examples of discrete random variables:
1. Number of hearts drawn from a deck of playing cards.
2. Number of heads in 3 tosses of a fair coin.
3. Number of persons in a city objecting to a new ordinance
Some examples of continuous random variable:
1. Weight of persons
2. Height of a person
3. Percentage of the students who passed the exam
4. Income/Money
Can you classify me?
Directions: Classify the following random variables as discrete or continuous.
1. The number of defective computers produced by a manufacturer in a
certain school
2. The weight of newborns each year in a hospital
3. The number of siblings in a family of a region
4. The amount of paint utilized in a classroom building project of ICNHS
Activity 4.

5. The number of dropout in a school district for a period of 10 years
6. The speed of a car
7. The number of female athletes of the Sports track
8. The time needed to finish the test
9. The amount of sugar in a cup of coffee
10.The number of people who are playing LOTTO each day
11.The number of accidents per year at an intersection
12.The number of voters favoring a candidate
13.The number of patient arrivals per hour at a medical clinic
14.The average amount of electricity consumed per household per month
15.The number of deaths per year attributed to lung cancer
LESSON 2: DISCRETE PROBABILITY DISTRIBUTION
CODE: M11/12SP-IIIa-4, M11/12SP-IIIa-5 and M11/12SP-IIIa-6
OBJECTIVES:
In this lesson, the students will be able to define and illustrate a probability
distribution for a discrete random variable and its properties, construct the
probability mass function of a discrete random variable and its corresponding
histogram and compute probabilities corresponding to a given random
variable.
DISCUSSION:
A discrete random variable has a corresponding probability distribution
that is called to be a discrete probability distribution.
What is my probability?
Directions: Find the probability of the following events.
Event (E) Probability P(E)
Activity 1.

1. Getting an even number in a single roll of a
die
2. Getting a sum of 6 when two diced are rolled
3. Getting an ace when a card is drawn from a
deck
4. The probability that all children are boys if a
couple has three children
5. Getting an odd number and a tail when a die
is rolled and a coin is tossed simultaneously
6. Getting a sum of 11 when two dice are rolled
7. Getting a black card and a 10 when a card is
drawn from a deck
8. Getting a red queen when a card is drawn
from a deck
9. Getting doubles when two dice are rolled
10.Getting a red ball from a box containing 3 red
and 6 black balls
Definition (Discrete Probability Distribution)
A discrete probability distribution is an equation, or a table that lists
all possible values that a discrete random variable can take on together with
associated probabilities.
Note: A histogram is a bar graph. To construct a histogram for a probability
distribution, follow these steps:
 Plot the values of the random variable along the horizontal axis.
 Plot the probabilities along the vertical axis.
Examples:
1. Construct a probability distribution or the probability mass function of
discrete random variable X ; X is the number of heads when a coin is
tossed thrice. Make a histogram for this probability distribution.

Solution: Constructing a probability distribution or the probability mass
function of discrete random variable X can be done in two ways: the
tabular form and the equation form. For the tabular form, it includes all
the possible outcomes, the random variable and the probability i.e.
Outcomes TTT TTH THT THH HTT HTH HHT HHH

X number of heads 0 1 1 2 1 2 2 3
Thus,
heads
of
number
X  0 1 2 3
 
X
P 8
1 8
3 8
3 8
1
For the equation form,  












3
,
8
1
2
,
8
3
1
,
8
3
0
,
8
1
x
if
x
if
x
if
x
if
X
P . The histogram for this
probability distribution is illustrated below:
2. List the possible values of the following random variables. Find the
probability distribution or the probability mass function of discrete random
variable. Make a histogram for each probability distribution below.
a. Let Y be the random variable denoting the sum of the points on the
upturned faces of the dice when a pair of dice is rolled.
0
0
0
1/7
1/5
1/4
2/7
1/3
2/5
0 1 2 3
PROBABILITY
p(x)
NUMBER OF HEADS

Solution: Using the tabular form, the probability distribution or the
probability mass function of the discrete random variable Y is

Y sum of
the points
on the
upturned
faces
2 3 4 5 6 7 8 9 10 11 12
 
Y
P 36
1 18
1 12
1
9
1
36
5 1/2 36
5
9
1 12
1 18
1 36
1
Below is the histogram of the table:
b. Let Z be the random variable denoting the number of heads when a coin is
tossed five times.
Solution: The probability distribution or the probability mass function of
the discrete random variable Z is

Z number of
heads
5 4 3 2 1 0
 
Z
P 32
1 32
5 16
5 16
5 32
5 32
1
0
1/50
1/25
3/50
2/25
1/10
3/25
7/50
4/25
9/50
2 3 4 5 6 7 8 9 10 11 12
P(Y)
SUM OF THE POINTS

Below is the histogram of the table:
Remarks: (Properties of a Probability Distribution)
1. The values of a probability distribution are probabilities; they must be
numbers on the interval from 0 to 1. In symbol, we write it as   1
0 
 x
P
.
2. The sum of all the values of a probability distribution must be equal to 1.
In symbol, we write it as   1

 x
P .
Can you justify me?
Directions: Determine whether the distribution represents a probability
distribution. Justify your answer.
1. X 1 5 8 7 4. X 4 8 12 15 17
P(X) 3
1
3
1
3
1
3
1
P(X) 5
1
8
1
8
1
5
1
8
1
2. X 0 2 4 6 5. X 1 3 5 7
P(X) 6
1
6
1
3
1
6
1
P(X) 0.35 0.25 0.22 0.12
3. X 1 2 3 5
0
1/20
1/10
3/20
1/5
1/4
3/10
7/20
5 4 3 2 1 0
P(Z)
NUMBER OF HEADS
Activity 2.

P(X)
4
1
8
1
4
1
8
1
Can you construct me?
Directions: Construct the probability distribution for the random variables
described in each of the following situations. Draw the corresponding histogram
for each probability distribution.
1. Suppose three cell phones are tested at random where D represent
the defective cell phone and N represent the non-defective cell
phone. Let X represents the number of defective cell phones that
occur. , show the values of the random variable X.
2. Two balls are drawn in succession without replacement from an urn
containing 5 red balls and 6 blue balls. Let Z be the random variable
representing the number of blue balls. Find the values of the random
variable Z.
3. Four coins are tossed. Let Y be the random variable representing the
number of heads that occur. Find the values of the random variable
Y.
4. A shipment of five computers contains that are slightly defective. If a
retailer receives three of these computers at random, list the
elements of the sample space S using the letters D and N for
defective and non-defective computers, respectively. To each
sample point, assign a value x of the random variable X representing
the number of computers purchased by the retailer which are slightly
defective.
Activity 3.

A random variable is a numerical description of the outcome of a
statistical experiment. A random variable that may assume only a finite number
or an infinite sequence of values is said to be discrete; one that may assume
any value in some interval on the real number line is said to be continuous. For
instance, a random variable representing the number of automobiles sold at a
particular dealership on one day would be discrete, while a random variable
representing the weight of a person in kilograms (or pounds) would be
continuous.
The probability distribution for a random variable describes how the
probabilities are distributed over the values of the random variable. For a
discrete random variable, x, the probability distribution is defined by a
probability mass function, denoted by f(x). This function provides the probability
for each value of the random variable. In the development of the probability
function for a discrete random variable, two conditions must be satisfied: (1) f(x)
SUMMARY.

must be nonnegative for each value of the random variable, and (2) the sum of
the probabilities for each value of the random variable must equal one.
Multiple Choice: Choose the letter that you think the best
answer to each of the following questions. Write the letter
of your choice on the space provided.
______ 1. The sample space of rolling a die and tossing a coin simultaneously.
a.  
T
H
T
H
T
H
S 6
,
5
,
4
,
3
,
2
,
1

b.  
T
H
T
H
T
H
T
H
T
H
T
H
S 6
,
6
,
5
,
5
,
4
,
4
,
3
,
3
,
2
,
2
,
1
,
1
 answer
c.  
T
T
T
H
H
H
S 6
,
5
,
4
,
3
,
2
,
1

d.  
,
6
,
5
,
4
,
3
,
2
,
1 H
H
H
T
T
T
S 
______ 2. A pair of dice is rolled. Let Y be the random variable denoting the
sum of the points on the upturned faces of the dice. Find the values of the
random variable Y.
a. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 c. 1, 3, 5, 7, 9 and 11
b. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 d. 2, 4, 6, 8, 10 and 12
______ 3. Which of the following is a discrete random variable?
a. the average amount of electricity consumed
b. the number of persons eating seafood in a restaurant
c. the percentage of the students who passed the exam
d. the average weight of female athletes
______ 4. The following are continuous random variable except _______.
a. weight of persons c. income/money
b. height of a person d. number of heads in 3 tosses of a fair coin
______ 5. If a pair of dice is rolled, which is not a possible value of the random
variable for the sum of the points on the upturned faces of the dice?
a. 1 b. 4 c. 7 d. 11
______ 6. What makes the distribution below a probability distribution?
POST-TEST.

a. The random variable is equally distributed.
b. The sum of all values of P(x) is equal to one.
c. The values of a probability distribution are probabilities.
d. The values of the P(x) are numbers on the interval from 0 to 1.
______ 7. A function whose value is a real number determined by each element
in the sample space – the set of all possible outcomes of an experiment.
a. random variable c. density function
b. probability distribution d. histogram
______ 8. Which formula gives the probability distribution shown by the table?
X 1 3 5
P(X) 6
1
3
1
2
1
a.  
12
1


X
X
P b.  
6
X
X
P  c.  
18
2


X
X
P d.  
6
1

X
P
______ 9. Four coins are tossed. Let Y be the random variable representing
the number of heads that occur. The probability distribution of the random
variable Y is _______.
a.
b.
c.
d.
______ 10. If  
6
X
X
P  , what are the possible values of X for it to be a
probability distribution?
a. 1, 2, 3 b. 0, 2, 3 c. 2, 3, 4 d. 1, 1, 2
X 0 2 4 6 8
P(X) 6
1
6
1
3
1
6
1
6
1
𝑦 = 𝑜𝑓 ℎ𝑒𝑎𝑑𝑠 0 1 2 3 4
P(x) 4/16 6/16 1/16 4/16 1/16
𝑦 = 𝑜𝑓 ℎ𝑒𝑎𝑑𝑠 0 1 2 3 4
P(x) 1/16 4/16 6/16 4/16 1/16
𝑦 = 𝑜𝑓 ℎ𝑒𝑎𝑑𝑠 0 1 2 3 4
P(x) 6/16 4/16 1/16 4/16 1/16
𝑦 = 𝑜𝑓 ℎ𝑒𝑎𝑑𝑠 0 1 2 3 4
P(x) 1/16 4/16 1/16 4/16 6/16

POST-TEST
1. B
2. A
3. B
4. D
5. A
6. B
7. A
8. A
9. B
10. A
ANSWER KEY.

References:
Source: Random Variables and Probability Distributions. Encyclopaedia
Britannica, Inc., last modified February 25, 2019
https://www.britannica.com/science/statistics/Random-variables-and
probability-distributions
Source: Random Variables, Distributions and Expected Value. Professor Paul
Glasserman, last modified 2001.
https://www0.gsb.columbia.edu/faculty/pglasserman/B6014/RandomVariables
.pdf
Source: Probability Distributions for Discrete Random Variables. Powered by
MindTouch® and are supported by the Department of Education Open
Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis
Library, the California State University Affordable Learning Solutions Program,
and Merlot, last modified June 6, 2019
https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Intro
ductory_Statistics_(Shafer_and_Zhang)/04%3A_Discrete_Random_Variables
/4.2%3A_Probability_Distributions_for_Discrete_Random_Variables

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