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Statistics and Probability
Quarter 3 – Module 1
Random Variables and Probability
Distribution
Learning Competencies:
Illustrates a random variable (discrete and continuous).
M11/12SP-IIIa-1
Distinguishes between a discrete and continuous random
variable. M11/12SP-IIIa-2
Finds the possible values of random variable.
M11/12SP-IIIa-3
Illustrates a probability distribution for a discrete random
variable and its properties. M11/12SP-IIIa-4
Computes probabilities corresponding to a given random
variable M11/12SP-IIIa-6
At the end of the lesson, you are expected to:
 illustrate a random variable;
 distinguish between a discrete and continuous random variable;
 find the possible values of random variable; and
 illustrate a probability distribution for a discrete random variable and its
properties; and
 compute probabilities corresponding to a given random variable.
Lesson 1: Random Variables (Discrete and Continuous)
What is it…
A random variable is unknown value or a function that assigns values to each of
an experiment's outcomes. Random variables are often designated by letters and can be
classified as discrete, which are variables that have specific values, or continuous,
which are variables that can have any values within a continuous range.
Random variables are often used in econometric or regression analysis to
determine statistical relationships among one another.
In Statistics and Probability, random variables are used to quantify outcomes of
a random occurrence, and therefore, can take on many values. Random variables are
required to be measurable and are typically real numbers. For example, the letter X may
be designated to represent the sum of the resulting numbers after three dice are rolled.
In this case, X could be 3 (1 + 1+ 1), 18 (6 + 6 + 6), or somewhere between 3 and 18,
since the highest number of a die is 6 and the lowest number is 1.
A random variable is different from an algebraic variable. The variable in an
algebraic equation is an unknown value that can be calculated. The equation 10+x =13
shows that we can calculate the specific value for x which is 3. On the other hand, a
random variable has a set of values, and any of those values could be the resulting
outcome as seen in the example of the dice above.
In the corporate world, random variables can be assigned to properties such as
the average price of an asset over a given time period, the return on investment after a
specified number of years, the estimated turnover rate at a company within the following
six months, etc.
There are two types of random variables, the discrete and continuous random
variables.
Types of Random Variables
Discrete
Continuous
Has infinite numerical values
associated with any interval
on the number line system
without any gaps or breaks.
Is
a
numerical
value
associated with the desired
outcomes. It is also either a
finite or infinite number of
values but countable such as
whole numbers 0,1,2,3.
What’s More…
A. Answer the question.
What are random variables?
B. Classify the following random variables as DISCRETE or CONTINUOUS. In the
answer sheet, write D if your answer is Discrete and C if your answer is
Continuous.
1. number of siblings in a family
2. weight of newborns each year in a hospital
3. number of defective computers produced by a manufacturer
4. speed of a car
5. time needed to finish the test
6. number of female athletes
7. average amount of electricity consumed per household per month
8. number of voters favoring a candidate
9. number of dropouts in a school
10. number of patient’s arrival per day at a medical clinic
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Lesson 2: Possible Values of Random Variable
What is it…
Recall that a variable is a characteristics or attribute that can assume different
values. We use capital letters to denote or represent a variable. In this lesson, we shall
discuss variables that are associated with probabilities, called random variables.
Suppose three cell phones are tested at random. We want to find out the
number of defective cell phones that occur. Thus, to each outcome in the sample
space we assign a value. These are 0, 1, 2, or 3. If there is no defective cell phone,
we assign the number 0; if there is one defective cell phone, we assign the number
1; if there are two defective cell phone, we assign the number 2; and 3, if there are
three defective cell phone. The number of defective cell phones is a random variable.
The possible values of this random variables are 0, 1, 2 and 3.
Illustration:
Let D represent the defective cell phone and N represent the non-defective cell
phone. If X be the random variable representing the number of defective cell phones,
show the values of the random variable X. The table should look like this.
Possible Outcomes
NNN
NND
NDN
DNN
NDD
DND
DDN
DDD
Value of the Random Variable X
(number of defective cell phones)
0
1
1
1
2
2
2
3
So, the possible values of the random variable X are 0, 1, 2, and 3.
What’s More…
Directions: Solve the given problem. Show your solution.
Four coins are tossed. Let Z be the random variable representing the number
of heads that occur. Find the values of the random variable Z.
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Lesson 3: Discrete Probability Distribution
What is it…
A discrete probability distribution or a probability mass function consists of values
a random variable can assume and the corresponding probabilities of the values.
Properties of Probability Distributions of Discrete Random Variables
1. The probability of each value of the random variable must be between or
equal to 0 and 1. In 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 symbol, we write it as Σ P(X) = 1.
Example:
Table 1. The Probability Distribution of the Probability Mass Function of
Discrete Random Variable X
Number of Blue Balls X
0
1
2
Probability P(X)
1⁄
4
1⁄
2
1⁄
4
This represents a probability distribution since it satisfies the two properties. The
probability of each value is between 0 and 1 ( 1⁄4 and 1⁄2). Moreover, the sum of all
values of the probabilities is equal to one. 1⁄4 + 1⁄2 + 1⁄4 = 1.
The probability distribution of a discrete random variable X is a list of each
possible value of X together with the probability that X takes that value in one trial of
the experiment.
Example:
Suppose three coins are tossed. Let Y be the random variable representing the
number of tails that occur. Find the probability of each of the values of the random
variable Y.
Solution:
STEPS
1. Determine the sample space. Let H
represent head and T represent tail.
Solution
The sample space for this experiment
is
S = {TTT, TTH, THT, HTT, HHT, HTH,
THH, HHH}
2. Count the number of tails in each outcome
Possible
Value of the
in the sample space and assign this number to
outcomes
random variable
this outcome.
Y
(number of tails)
TTT
3
TTH
2
THT
2
HTT
2
HHT
1
HTH
1
THH
1
HHH
0
4
3. There are four possible values of the random
variable Y representing the number of tails.
These are 0, 1, 2 and 3. Assign probability
values P(Y) to each value of the random variable.
 There are 8 possible outcomes and no tail
occurs once, so the probability that we shall
assign to the random variable 0 is 𝟏⁄𝟖
 There are 8 possible outcomes and 1 tail
occurs three times, so the probability that we
shall assign to the random variable 1 is 𝟑⁄𝟖
 There are 8 possible outcomes and 2 tails
occur three times, so the probability that we
shall assign to the random variable 2 is 𝟑⁄𝟖
There are 8 possible outcomes and 3 tails
occur once, so the probability that we shall
assign to the random variable 3 is 𝟏⁄𝟖
Number of
Tails
Y
Probability
P(Y)
𝟏⁄
𝟖
𝟑⁄
𝟖
𝟑⁄
𝟖
𝟏⁄
𝟖
0
1
2
3
The Probability Distribution on the Probability Mass
Function of Discrete Random Variable Y
Number of Tails Y
Probability P(Y)
0
𝟏⁄
𝟖
1
𝟑⁄
𝟖
2
𝟑⁄
𝟖
3
𝟏⁄
𝟖
Histogram
Construct a histogram for this probability distribution. 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.
In plotting the probabilities along the vertical axis, you can change fractions to
decimals (ex. 1⁄8 = 0.125, 1⁄4 = 0.25 and 1⁄2= 0.5)
The Histogram for the Probability Distribution
of the Discrete Random Variable Y
0.40
0.35
Probability P(Y)


0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
1
2
Number of Tails Y
5
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What’s More...
A. Determine whether the distribution represents a probability distribution. Explain
your answer.
1.
X
P(X)
2.
X
P(X)
1
1⁄
3
5
1⁄
3
8
1⁄
3
7
1⁄
3
9
1⁄
3
0
1⁄
6
2
1⁄
6
4
1⁄
3
6
1⁄
6
8
1⁄
6
B. Construct the probability distribution of the random variable Z.
Two ball 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.
Assessment
Directions: Choose the letter that corresponds to the correct answer. Write your
answer in the answer sheet provided.
1. Which of the following best describe variable that can be counted?
A. Discrete
B. Measured
C. Nominal
D. Qualitative
2. Which of the following best describe variable that can be measured?
A. Discrete
B. Continuous
C. Nominal
D. Qualitative
3. Which of the following is a NOT continuous variable?
A. A person’s weight each year
B. A person’s height on each birthday
C. Number of bicycle finished in a factory each day
D. The amount of water in a pale
4. Which of the following is NOT a discrete variable?
A. The number of students present in a class
B. The number of death per year attributed to kidney failure
C. The average amount of water consumed per household per month
D. The number of patients in a hospital each day
5. Which of the following is NOT a true statement?
A. The value of a 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 the probabilities in a probability distribution is always equal
to one.
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6. If two coins are tossed, which is not a possible value of the random variable for the
number of tails?
A. 0
B. 1
C. 2
D. 3
7. Which formula gives the probability distribution shown by the table?
X
P(X)
A. P(X)= 1⁄X
B. P(X)= X⁄6
2
1⁄
2
3
1⁄
3
6
1⁄
6
C. P(X)= 6⁄X
D. P(X)= 1⁄6
8. Which of the following cannot be the value of probability of the random variable?
A. 1.01
B. 0.3
C. 1/4
D. ½
9. In a local community, a couples were asked the questions “Are you satisfied with the
work of the current president?” If the husband and the wife both said “yes”, the response
is written as YY. If the husband said yes and the wife said “no”, the response is YN. Let
X = the number of “yes” responses, what are the possible values of the random
variables?
A. 0, 1, 2
C. 2, 3, 4
B. 1, 2, 3
D. 1, 1, 2
10. A set of numerical values assigned to a sample space is called
A. Random Experiment
C. Random Variable
B. Random Sample
D. Random process
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Answer Sheet
Name:
Grade & Section:
Score:
Quarter 3 – Module 1 (Statistics and Probability)
Lesson 1
What’s More
A.
1.
B.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Lesson 2
What’s More
Possible Outcomes
Value of the Random Variable Z
(number of heads)
Lesson 3
What’s More
A. 1.
2.
B.
Number of Blue Balls (Z)
Probability P(Z)
Assessment
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
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Quarter 3 - Module 1
Lesson 1
What’s More
A.
A random variable is a variable whose value is unknown or a function that
assigns values to each of an experiment's outcomes.
B.
1.
2.
3.
4.
D
C
D
C
5.
6.
7.
8.
C
D
C
D
9. D
10. D
Lesson 2
What’s More
Value of the Random Variable Z
(number of heads)
4
3
3
2
3
2
2
1
3
2
2
1
2
1
1
0
Possible Outcomes
HHHH
HHHT
HHTH
HHTT
HTHH
HTHT
HTTH
HTTT
THHH
THHT
THTH
THTT
TTHH
TTHT
TTTH
TTTT
Lesson 2
What’s More
A.
1. This distribution does not represent a probability distribution because the
sum of the probabilities of all values of the random variable is not equal to 1.
2. This represents a probability distribution because its value of the random
variables is between 0 and 1 and the sum of the probabilities of all values of the
random variable is equal to 1.
B.
Number of Blue Balls Z
Probability P(Z)
0
𝟏⁄
𝟒
1
𝟏⁄
𝟐
2
𝟏⁄
𝟒
Assessment
1.
2.
3.
4.
5.
6. D
7. A
8. B
9. A
10. C
A
B
C
C
B
Answer Key
Reference
Belecina, R. Statistics and Probability. Manila: Rex Book Store, Incorporated.
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