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1. Republic of the Philippines
Department of Education
Regional Office IX, Zamboanga Peninsula
Statistics and Probability
Quarter 3 – Module 1:
Discrete and Continuous Random Variables
Zest for Progress
Zeal of Partnership
11
Name of Learner: ___________________________
Grade & Section: ___________________________
Name of School: ___________________________

2. 1
What I Need to Know
The module contains only one lesson:
 Lesson 1 – Discrete and Continuous Random Variables
At the end of this module, you are expected to:
1. Illustrate a random variable (discrete or continuous);
2. Distinguishes between a discrete and a continuous random variable;
3. Finds the possible values of a random variable; and
4. Illustrates a probability distribution for a discrete random variable and its
properties.
What I Know
Directions: Encircle the letter of the correct answer.
1. Which is a discrete random variable?
A. The average amount of electricity consumed.
B. The number of patients in the hospital.
C. The amount of paint used in repainting a building.
D. The average weight of female athletes.
2. What is X in the equation 3+X = 2?
A. Solutions
B. The multiplier
C. Unknown number
D. Mathematical symbol
3. What are the possible outcomes to get Heads/Tails when flipping a coin?
A. 1 outcomes
B. 2 outcomes
C. 4 outcomes
D. 6 outcomes
4. What type of variable whose value is obtained by counting?
A. Continuous variable
B. Discrete variable
C. Quantitative variable
D. Qualitative variable

3. 2
5. What type of variable whose value is obtained by measuring?
A. Continuous
B. Discrete
C. Quantitative variable
D. Qualitative variable
6. What is the probability showing 4 dots when you consider a dice with the
property that the probability of a face with n dots showing up is
proportional to n?
A. ¼
B. 2/42
C. 3 ½
D. 4/21
7. Let X be a random variable with the probability distribution function f(x)
= 0.2 for |x|<1=0.1 for <|x|<4=0. What is the probability of P(0.5<x<5)?
A. 0.3
B. 0.4
C. 0.5
D. 0.8
8. What is the probability that tails turn up to 3 cases if the coin tossed up
to 4 times?
A. 1/6
B. 1/4
C. 1/3
D. 1/2
9. In a recent little league in softball game, each player went to bat 4 times.
The number of hits made by each player is described by the following
probability distribution.
Number of hits, X Probability P(X)
0 0.10
1 0.20
2 0.30
3 0.25
4 0.15
What is the mean of the probability distribution?
A. 1.00
B. 1.75
C. 2.00
D. 2.15
10. What is the mean if the probability of hitting the target is 0.40?
A. 0.2
B. 0.4
C. 0.6
D. 0.8

4. 3
What’s In
ACTIVITY 1: Toss Me!
Direction: Find the possible outcomes in tossing three coins. Complete the table below.
Possible Outcome
TTT
THT
HHH
What’s New
ACTIVITY 2: Pair Me Up!
Direction: List all the Possible Outcomes with the given problem below and find the
values of the random variable G.
Possible Outcomes
Value of Random Variable G
( number of green balls)
GG 2
From a box containing 2 green balls and 3 blue balls are drawn in
succession. Each ball is placed in the box before the next draw is made. Let G
be a random variable representing the number of green balls that occur. Find
the values of the random variable G.

5. 4
VARIABLES
QUALITATIVE
 (Also called
numerical)
data, whose
sizes are
meaningful,
answers
questions such
as “how many”.
QUANTITATIVE
 (Also called a
categorical
variable), are
variables that are
not numerical.
Discrete
 Are those
data that
can be
counted.
Continuous
 Are those
data that
can be
measured.
What is it
Random Variable is a variable whose possible values are determined by chance.
 Typically represented by an uppercase letter, usually X, while its corresponding
lowercase letter in this case, x, and is used to represent one of its values.
 For example: A coin is tossed thrice.
 Let the variable X represent the number of heads that result from this experiment.
1st toss 2nd toss 3rd toss Final No. of Heads(X)
Outcome
H HHH 3
H H
T HHT 2
H HTH 2
T
T HTT 1
H THH 2
T H T THT 1
T H TTH 1
T TTT 0

6. 5
In the illustration above, random variable is represented by the upper case X.
The lower case x represents the specific values.
Hence, x=3, x=2,x=1, x=2, x=1, x=1, and x=0.
The sample space for the possible outcomes is
S= { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
The value of the variable X can be 0, 1, 2, or 3. Then, in this example, X is a
random variable.
Random variables can either be discrete or continuous.
A discrete random variable can only take a finite (countable) number of distinct
values. Distinct values mean values that are exact and can be represented by
nonnegative whole numbers.
The following are examples of discrete random variables.
A. Let X = number of students randomly selected to be interviewed by a
researcher. This is a discrete random variable because its possible values are
0, 1, or 2, and so on.
B. Let Y = number of left-handed teachers randomly selected in a faculty room.
This is a discrete random variable because its possible values are 0, 1,or 2, and
so on.
Example 2: Let’s assume a test has five parts. We can define a discrete random
variables as X= # parts passed. So what are our possible outcomes?
Well X= 0,1,2,3,4, or 5
Note: A student can pass any number of parts (including zero) from this finite list.
CONTINUOUS RANDOM VARIABLE
 Can assume an infinite number of values in an interval between two specific
values. This means they can assume values that can be represented not only
by nonnegative whole numbers but also by fractions and decimals.
 These values are often results of measurements.
The following are example of continuous random variables:
1. Let Y = the weights of randomly selected students in pounds. This is a
continuous random variable because its values can be between any two
given weights. Also, weights are measured using s weighing scale. The
weight of a student for example can be 150.5lb but due to limits of
measuring devices, the measurement is always an approximate. The
weights of students can range from 100 to 180lb including all the decimal
places that come between these values.
A Discrete probability distribution or a probability mass function consists
of the values a random variable can assume and the corresponding
probabilities of the values.
 Can be represented in tubular, graphical, or formula form.

7. 6
Properties of a Probability Distribution
1. The probability of each value of the random variable must be between or
equal to 0 and 1. In the symbol, we write it as 0≤P(X)≤1.
2. The sum of the probabilities of all values of the random variable must
be equal to 1. In symbols, we write it as ∑P(X) = 1
3. Formula for the average / the mean (µ)
µ = ∑X . P(X)
Example 1: Consider the following table. Compute the Mean.
X P(X) X.P(X)
0 0.2 0
1 0.3 0.3
2 0.3 0.6
3 0.2 0.6
∑P(X) = 1 µ = ∑X . P(X)
mean or µ = 1.5
In the table, the random variable X assumes the values 0, 1 , 2 , and 3.
The corresponding probabilities of these values are 0.2, 0.3, 0.3, and 0.2, respectively.
These corresponding probabilities are each less than 1 but greater than 0 and when
added, the sum is 1.
∑P(X) = 0.2 +0.3 +0.3 +0.2
∑P(X) = 1
The mean of the discrete random variable X.P(x) is equal to µ = 1.5.
Example 2: Toss a fair of coin twice and let X be equal to the number of Heads (H)
observed.
Construct the discrete probability distribution of X.
P(X) = {HH, HT, TH, TT}
P(2) = ¼ or 0.25
P(1) = 2/4 or 0.50
P(0) = ¼ or 0.25
X 0 1 2
P(X) 1/4 1/2 1/4
The sum of all the probabilities is 1 so, the second property is also met.
∑P(X) = P(0) + P(1) + P(2)
= ¼ + ½ + ¼
∑P(X) = 1
Therefore, the distribution is a discrete probability distribution.

8. 7
What’s More
Activity 3.1: Identify Me!
Directions: Identify whether the given variable is discrete or continuous. Write D if it is
discrete and C if it is continuous. Write your answer on the blank space before each
item.
____________ 1. The number of books in the library.
____________ 2. The lifetime in hours of 15 flashlights.
____________ 3. The number of tourists each day in museum.
____________ 4. The capacity of water dams in a region.
____________ 5. The weight of Grade 1 pupils.
Activity 3.2: True or False
Directions: Write T if the statement is a discrete variable and write F if the statement is
a continuous variable. Write your answer on the blank space before each item.
___________ 1. The number of students present in the class.
___________ 2. Time it takes to get to school.
___________ 3. The distance travelled between classes.
___________ 4. The number of red marbles in a jar.
___________ 5. The number of heads when flipping three coins.
Activity 3.3: Identify Me!
Directions: Identify discrete probability distributions? If it is not a discrete probability
distribution, identify the property or properties that are not satisfied.
A.
x 1 2 3 4 5
P(x) 0.10 0.20 0.25 0.40 0.50
B.
x 1 2 3 4 5
P(x) 0.05 0.25 0.34 0.28 0.08
C.
x 1 2 3 4 5
P(x) 0.08 0.1 0.34 0.31 0.26
D.
x 1 2 3 4 5
P(x) 0.03 0.22 0.5 0.23 0.02
E.
x 1 2 3 4 5
P(x) 0.05 0.27 0.34 0.28 0.06

9. 8
RANDOM VARIABLES DISCRETE PROBABILITY DISTRIBUTION
CONTINUOUS RANDOM VARIABLE DISCRETE RANDOM VARIABLE
What I Have Learned
Activity 4.1: FLIP ME, NOW!
Direction: Fill in the blanks. Use the highlighted words to answer the questions below.
Write your answers on a separate sheet of paper.
___________ 1. A probability mass function consists of the values a random
variable can assume and the corresponding probabilities of the values.
___________ 2. A variable whose possible values are determined by chance and
typically represented by an uppercase letter usually X.
___________ 3. A type of variable whose values are often the results of
measurement.
___________ 4. A type of variable that can only take a finite (countable) number of
distinct values.
ACTIVITY 4.2: LOOK BACK AND REFLECT
______________________________________________________________________________
Questions:
1. How do you find the values of a random variable?

2. How do you know whether a random variable is continuous or discrete?

3. What is the difference between continuous and discrete random variables?
What I Can Do
ACTIVITY 5: SHOW ME WHAT YOU GOT!
Direction: Problem Solving. Find the following:
A. Find the probability that the arrow will stop at 1, 2, 3, and 4.
P(1) =
P(2) =

10. 9
P(3) =
P(4) =
B. Construct the discrete probability distribution of the random variable X.
1 2
3 4
Assessment
Direction: Choose the correct letter of your answer. Write the letter on a separate sheet
of paper.
1. What is the possible outcome when rolling a die?
A. 2 B. 3 C. 4 D.
2. How do you describe a discrete random variable?
A. The set of all possible outcomes of an experiment.
B. Often represents measured data, such as heights, weights, and temperature.
C. Mostly represent count data, such as the number of defective chairs produced
in a factory.
D. It is a variable whose values are determined by chance.
3. Which is NOT a discrete random variable?
A. The number of tattoos that a randomly selected person has.
B. The height of a randomly selected student in a group.
C. The number of grammatical errors in a 300-word essay composition of a senior
high school student.
D. A student’s score in a true-or false quiz consisting of questions.
4. Which is a discrete random variable?
A. The weights of mechanically produced items.
B. The number of children at a Christmas Party.
C. The times in seconds, from a 100m sprint.
D. The distance between Centre Point Tower.
5. Which is NOT a true statement?
A. The value of the random variable could be zero.
B. Random variables can only have one value.
C. The probability of the value of a random variable could be zero.
D. The sum of all probabilities in a probability distribution is always equal to 1.
The spinner below is divided into four sections. Let X be the score where
the arrow will stop (numbered as 1, 2, 3, 4, in the drawing below).

11. 10
6. You decide to collect a bunch of cans of soda and measure the volume of soda in
each can. Let X = the number of mL of soda in each can. What type of variable is
X?
A. X is a discrete random variable.
B. X is a continuous random variable.
C. X is a not continuous random variable.
D. X is not a random variable.
7. Which is NOT a property of random variable?
A. The sum of a random probability of a random variable is equal to 1.
B. A random variable cannot be negative.
C. A random variable represents numerical outcomes for different situations or
events.
D. A random variable can be discrete or continuous.
8. If the probability that a bomb dropped from a place will strike the target 60% and
if 10 bombs are dropped. What is the mean?
A. 0.4 B. 0.6 C. 4 D. 6
9. Which is a probability distribution?
A. The sum of the probabilities of all values of the random variable must be equal
to 1.
B. The sum of the probabilities of all values of the random variable must be less
than 1.
C. The sum of the probabilities of all values of the random variable must be equal
to zero.
D. The sum of the probabilities of all values of the random variable must be more
than 1.
10. What is the mean in tossing 8 coins?
A. 1 B. 2 C. 4 D. 8
Additional Activities
ACTIVITY 6: COMPLETE THE MISSING ME!
Direction: Complete the table below and find the mean of the following probability
distribution.
X P(X) X.P(X)
1 1/7
6 1/7
11 3/7
16 1/7
21 1/7
Total 1

12. 11
References
Albert, J. R. G. (2008), Basic Statistics for the Tertiary Level ( ed. Roberto Padua,
Welfredo Patungan, Nelia Marquez), published by Rex Store.
Handbook of Statistics 1 (1st and 2nd Edition), Authored by the Faculty of the Institute of
Statistics, UP Los Banos, College Laguna 4031.
Mercado, J. P. (2016), Next Century Mathematics Grade 11 / Grade 12, published by
Phoenix Publishing House, Inc.
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Talusan National High School
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EPS in Mathematics
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