2. ● Illustrate the mean and variance of a discrete
random variable;
● Calculate the mean and the variance of a discrete
random variable;
● Interpret the mean and the variance of a
discrete random variable; and
● Solve problems involving mean and variance of
probability distributions.
OBJECTIVES
3. Mean of the Discrete
Random Variable
● Covid-19 is continuously spreading around
the world, that is why reports regarding
average infected people per country is
being updated every day. For this kind of
report, experts used Statistics and
Probability to show reliable analysis in
their data.
● In this lesson, you will learn how to
compute the average or mean of a
discrete probability distribution as well as
the variance and standard deviation of a
discrete random variable.
4. What’s New
Consider the outcomes of a coin tossed as a random event. The probability of getting tail
is ½ or 50%, and the probability of getting head is ½ or 50% also, but it is hard to predict
the outcome that will occur. In this lesson, you will learn how to determine the likeliHOOD
of the happening of an event.
Mean of a Discrete Random Variable
The Mean µ of a discrete random variable is the central value or
average of its corresponding probability mass function. It is also
called
as the Expected Value. It is computed using the formula:
µ = ∑x p(x)
Where x is the outcome and p(x) is the probability of the outcome.
5. What is It
Examples:
1. Determine the mean or Expected Value of random variable below.
Therefore, mean is 2 for the above random variable.
6. 2. Find the mean of the random variable Y representing the number of red color chocolates per
160-gram pack of colored chocolate packages that has the following probability distribution.
Solution
µ = ∑ ⟮ ( )⟯
= ∑ ⟮4(0.10) + 5(0.37) + 6(0.33) + 7(0.20)⟯
= ∑ ⟮0.40 + 1.85 + 1.98 + 1.40⟯
= 5.63
So, the mean of the probability distribution is 5.63. This implies that the average
number of red chocolates per 160-gram is 5.63.
Y 4 5 6 7
P(y) 0.10 0.37 0.33 0.20
7. 3. The probabilities that a customer will buy 1, 2, 3, 4, or 5 items in a grocery
store are
3
10,
1
10,
1
10,
2
10, and
3
10, respectively. What is the average number of items
that a customer will buy?
To solve the above problem, we will follow 3 steps below.
STEPS IN FINDING THE MEAN
Step 1: Construct the probability distribution for the random variable X
representing the number of items that the customer will buy.
Step 2: Multiply the value of the random variable X by the corresponding
probability.
Step 3: Add the results obtained in Step 2. Results obtained is the mean of the
probability distribution.
9. Solution Continuation
So, the mean of the probability distribution is 3.1. This implies that the average number of
items that the customer will buy is 3.1.
10. Variance and Standard Deviation of the
Discrete Random Variable
The variance and standard deviation describe the amount of
spread, dispersion, or variability of the items in a distribution.
How can we describe the spread or dispersion in a
probability distribution? In this lesson, you will learn how to
compute the variance and standard deviation of a discrete
probability distribution.
Now, let us find out how can we find the variance and
standard deviation of a discrete probability distribution.
11. What’s New
Variance and Standard Deviation of a Random Variable
The variance and standard deviation are two values that describe how
scattered or spread out the scores are from the mean value of the
random variable. The variance, denoted as σ2, is determined using the
formula:
σ2 = ∑( x − µ)2 p(x)
The standard deviation σ is the square root of the variance, thus,
σ = √ ∑( x − µ)² p (x )
σ2 - variance σ – standard deviation
µ - mean p(x) – probability of the outcome
12. What is It
Let’s try!
Let’s have examples:
1. The number of cars sold per day at a local car dealership, along with
its corresponding probabilities, is shown in the succeeding table.
Compute the variance and the standard deviation of the probability
distribution by following the given steps. Write your answer in your
answer sheets.
Number of Cars Sold X Probability P(x)
0 10%
1 20%
2 30%
3 20%
4 20%
13. In solving the problem, let’s follow the steps below.
STEPS IN FINDING THE VARIANCE AND STANDARD DEVIATION
1. Find the mean of the probability distribution.
2. Subtract the mean from each value of the random variable X.
3. Square the result obtained in Step 2.
4. Multiply the results obtained in Step 3 by the corresponding
probability.
5. Get the sum of the results obtained in Step 4. Results obtained is
the value of the variance of probability distribution.
16. To Solve for Standard Deviation:
Get the square root of the variance
σ2 = ∑( x− µ)2p(x)
= 1.56
σ = √1.56
= 1.25
So, the variance of the number of cars sold per day is 1.56 and
the standard deviation is 1.25.
17. 2. When three coins are tossed once, the probability distribution for the random variable X representing the
number of heads that occur is given below. Compute the variance and standard deviation of the probability
distribution.
Solution:
Follow the steps in finding variance and standard deviation of the probability
distribution.
18.
19.
20. To solve for Standard Deviation
σ2 = ∑(x − µ)2p(x)
= 0.74
σ = √0.74
= 0.86
The mean in tossing 3 coins with probability of Head will show
up is 0.86 and the variance is 0.74, then the standard deviation is
0.86.
21. What’s More
A. Determine the mean or expected value of each Random Variable. Write your answer in
your answer sheets.
1.
2.
3.
s 3 4 12 20
P(s) 0.1 0.5 0.2 0.2
t 5 10 20
P(t) 50% 12% 38%
w 1/12 1/6 1/3 1/2
P(w) 1/2 1/10 1/5 1/5
22. 4. Find the mean of the probability distribution of the random variable X, which can take
only the values 1, 2, and 3, given that P(1) =1033, P(2) = 13, and P(3) = 1233.
5. The probabilities of a machine manufacturing 0, 1, 2, 3, 4, and 5 defective parts in one
day are 0.75, 0.17, 0.04, 0.025, 0.01, and 0.005 respectively. Find the mean of the
probability distribution.
B. Determine the Variance and Standard Deviation of each random variable. Write your
answer in your answer sheets.
1. x 1 2 3 4 5
P(x) 1
5
1
5
1
5
1
5
1
5
23. 2
.
3
.
z 2 4 6 8
P(z) 0.6 0.1 0.2 0.1
m 1 3 5 7
P(m) 40% 25% 15% 20%
4. The random variable X, representing the number of nuts in a chocolate bar
has the following probability distribution. Compute the variance and standard
deviation.
24. 5. The number of items sold per day in a sari-sari store, with its corresponding
probabilities, is shown in the table below. Find the variance and standard
deviation of the probability distribution.
.
Number of Items Sold
X
Probability
P(x)
19 0.20
20 0.20
21 0.30
22 0.20
23 0.10
25. Answer the following questions in your own understanding.
1. How to compute the mean of a discrete random variable?
State the 3 steps. Write your answer in your answer sheets.
2. How to find the variance and standard deviation of a
discrete random variable? Write your answer in your
answer sheets.
What I Have Learned
26. What I Can Do
Make a study about how many sheets of paper you consumed weekly
in answering your Self Learning Modules. Record the quantity (total number
of sheets) per subject, then construct a probability distribution. Compute the
mean, variance, and the standard deviation of the probability distribution
you made. Interpret the result, then find out how many weeks you will
consume 50 sheets of pad paper.
27. Assessment
Find the mean, variance, and standard deviation of the following probability distribution
then interpret the computed values.
1. Variable z representing the number of male teachers per Elementary school.
2. The number of mobile phones sold per day at a retail store varies as shown in
the given probability distribution below. Find the expected number of mobile
phones that will be sold in one day.
z 2 3 4 5 6
P(z) 40% 32% 11% 9% 8%
x 30 33 38 40 50
P(x) 0.2 0.2 0.35 0.23 0.02
28. 3. Number of monthly absences of Juan Dela Cruz based on his previous records of
absences as presented in the probability distribution below.
Number of Absences (X) Percent P(x)
3 25%
4 30%
5 30%
6 15%
Number of Computer Sold X Probability P(x)
0 0.1
2 0.2
5 0.3
7 0.2
9 0.2
4. The number of computers sold per day at a local computer store, along with its
corresponding probabilities, is shown in the table below.
29. 5. The number of inquiries received per day by the office of Admission in SHS X
last enrolment is shown below.
Number of Inquiries X Probability P(x)
22 0.08
25 0.19
26 0.36
28 0.25
29 0.07
30 0.05
