Midterms for Statistics

1. LESSON 5:
CALCULATING MEAN AND
VARIANCE OF A DISCRETE
RANDOM VARIABLE

2. THE MEAN
a. The average of all possible outcomes. It is
otherwise referred to as the “expected value”
of a probability distribution.
b. Means that if we repeat any given experiment
infinite times, the theoretical mean would be the
“expected value”.
c. Any discrete probability distribution has a
mean.

3. THE VARIANCE AND STANDARD
DEVIATION
The VARIANCE of a random
variable displays the variability or the
dispersions of the random variables.
It shows the distance of a random
variable from its mean.

4. THE VARIANCE AND STANDARD
DEVIATION
a. If the values of the variance and standard deviation
are HIGH, that means that the individual outcomes
of the experiment are far relative to each other. In
other words, the values differ greatly.
b. A large value of standard deviation (or variance)
means that the distribution is spread out, with
some possibility of observing values at some
distance from the mean.

5. THE VARIANCE AND STANDARD
DEVIATION
c. If the variance and standard deviation are LOW, that
means that the individual outcomes of the
experiment are closely spaced with each other. In
other words, the values are almost the same values
or if they do differ, the difference is small.
d. A small value of standard deviation (or variance)
means that the dispersion of the random variable is
narrowly concentrated around the mean.

6. Mean, variance, and
standard deviation can
be illustrated by looking
pattern and analyzing given
illustrations and diagrams.

7. During Town Fiesta, people used to go to Carnival
that most folks call it “Perya”. Mang Ben used to play
“Beto–beto” hoping that he would win. While he is
thinking about what possible outcomes in every roll
would be, he is always hoping that his bet is right.
X 1 2 3 4 5 6
P(X) 𝟏
𝟔
𝟏
𝟔
𝟏
𝟔
𝟏
𝟔
𝟏
𝟔
𝟏
𝟔
EXAMPLE

8. 1.Is the probability of X lies between 0 and 1?
2.What is the sum of all probabilities of X?
3.Is there a negative probability? Is it possible
to have a negative probability?
4.How will you illustrate the average or mean
of the probabilities of discrete random
variable?
QUESTIONS:

9. 1.Is the probability of x lies between 0
and 1?
Yes, the probability of X lies between 0
and 1.
X 1 2 3 4 5 6
P(X) 𝟏
𝟔
𝟏
𝟔
𝟏
𝟔
𝟏
𝟔
𝟏
𝟔
𝟏
𝟔
SOLUTION:

10. 2. What is the sum of all probabilities of X?
෍ 𝑷 𝑿 = 𝑷 𝟏 + 𝑷 𝟐 + 𝑷 𝟑 + 𝑷 𝟒 + 𝑷 𝟓 + 𝑷 𝟔
=
𝟏
𝟔
+
𝟏
𝟔
+
𝟏
𝟔
+
𝟏
𝟔
+
𝟏
𝟔
+
𝟏
𝟔
=
𝟏+𝟏+𝟏+𝟏+𝟏+𝟏
𝟔
=
𝟔
𝟔
𝒐𝒓 𝟏
The sum of all probabilities of X is exactly 1.
X 1 2 3 4 5 6
P(X) 𝟏
𝟔
𝟏
𝟔
𝟏
𝟔
𝟏
𝟔
𝟏
𝟔
𝟏
𝟔
SOLUTION:

11. 3. Is there a negative probability? Is it possible
to have a negative probability?
No negative probabilities because it is
impossible to have it based on the
characteristic of the probability of
discrete random variables.
SOLUTION:

12. In a seafood restaurant, the
manager wants to know if their
customers like their new raw large
oysters. According to their sales
representative, in the past 4 months,
the number of oysters consumed by
a customer, along with its
corresponding probabilities, is
shown in the succeeding table.
Compute the mean, variance and
standard deviation.
Number of
oysters
consumed
X
Probabilit
y
P(X)
0 𝟐
𝟏𝟎
1 𝟐
𝟏𝟎
2 𝟑
𝟏𝟎
3 𝟐
𝟏𝟎
4 𝟏
EXAMPLE 1

13. (X) P(X) X ∗ P(X) 𝑋−𝜇. (𝑋 − 𝜇)² (𝑋 − 𝜇)² ∙ 𝑃(𝑋)
0 𝟎. 𝟐 𝟎 𝟎 − 𝟏. 𝟖 =
−𝟏. 𝟖
−𝟏. 𝟖 𝟐
= 𝟑. 𝟐𝟒
𝟑. 𝟐𝟒 ∗ 𝟎. 𝟐 = 𝟎. 𝟔𝟖𝟒
1 𝟎. 𝟐 0.2 𝟏 − 𝟏. 𝟖 = −𝟎. 𝟖 −𝟎. 𝟖 𝟐
= 𝟎. 𝟔𝟒
𝟎. 𝟔𝟒 ∗ 𝟎. 𝟐 = 𝟎. 𝟏𝟐𝟖
2 𝟎. 𝟑 0.6 𝟐 − 𝟏. 𝟖 = 𝟎. 𝟐 𝟎. 𝟐 𝟐
= 𝟎. 𝟎𝟒 𝟎. 𝟎𝟒 ∗ 𝟎. 𝟑 = 𝟎. 𝟎𝟏𝟐
3 𝟎. 𝟐 0.6 𝟑 − 𝟏. 𝟖 = 𝟏. 𝟐 𝟏. 𝟐 𝟐
= 𝟏. 𝟒𝟒 𝟏. 𝟒𝟒 ∗ 𝟎. 𝟐 = 𝟎. 𝟐𝟖𝟖
4 𝟎. 𝟏 0.4 4−𝟏. 𝟖 = 𝟐. 𝟐 𝟐. 𝟐 𝟐
= 𝟒. 𝟖𝟒 𝟒. 𝟖𝟒 ∗ 𝟎. 𝟏 = 𝟎. 𝟒𝟖𝟒

14. 1.What is the mean?
𝝁 = ෍[𝑿 ∗ 𝑷(𝑿)]
= 𝟎 + 𝟎. 𝟐 + 𝟎. 𝟔 + 𝟎. 𝟔 + 𝟎. 𝟒
𝝁 = 𝟏. 𝟖
SOLUTION:

15. 2. What is the standard deviation?
𝜎2
= ෍ (𝑋 − 𝜇)²∙ 𝑃(𝑋)
= 0.448 + 0.128 + 0.012 + 0.288 + 0.484
𝝈𝟐
𝟏. 𝟓𝟔
SOLUTION:

16. 3. What is the standard deviation?
𝜎 = σ (𝑋 − 𝜇)²∙ 𝑃(𝑋) 𝑂𝑅 𝜎 = 𝜎2
𝜎 = 1.56
𝝈 = 𝟏. 𝟐𝟓
SOLUTION:

17. Mr. Umali, a Mathematics
teacher, regularly gives a
formative assessment composed
of 5 multiple-choice items. After
the assessment, he used to
check the probability
distribution of the correct
responses, and the data is
presented below:
TEST ITEM
(X)
Probability
𝑷(𝑿)
0 0.03
1 0.05
2 0.12
3 0.30
4 0.28
5 0.22
EXAMPLE 2

18. 1.What is the average or mean of the given
probability distribution?
2.What are the values of the variance of the
probability distribution?
3.What are the values of standard deviation
of the probability distribution?
QUESTIONS:

19. X P(X) X ∗ P(X) X - 𝛍 (X - 𝛍)² (X - 𝛍)² ∗ P(X)
0 0.03 0 ∗ 0.03 = 0 0 − 3.41 = −3.41 (−3.41)² = 11.63 11.63 ∗ 𝟎. 𝟎𝟑 = 𝟎. 𝟑𝟓
1 0.05 1 ∗ 0.05
= 0.05
1 − 3.41 = −2.41 (−2.41)² = 5.81 5. 𝟖𝟏 ∗ 𝟎. 𝟎𝟓 = 𝟎. 𝟐𝟗
2 0.12 2 ∗ 0.12
= 0.24
2 − 3.41 = −1.41 (−1.41)² = 1.99 1. 𝟗𝟗 ∗ 𝟎. 𝟏𝟐 = 𝟎. 𝟐𝟒
3 0.30 3 ∗ 0.30
= 0.90
3 − 3.41 = −0.41 (−0.41)² = 0.17 0. 𝟏𝟕 ∗ 𝟎. 𝟑𝟎 = 𝟎. 𝟎𝟓
4 0.28 4 ∗ 0.28
= 1.12
4 − 3.41 = 0.59 (0.59)² = 0.35 0.35 ∗ 𝟎. 𝟐𝟖 = 𝟎. 𝟏𝟎
5 0.22 5 ∗ 0.22
= 1.10
5 − 3.41 = 1.59 (1.59)² = 2.53 2.53 ∗ 𝟎. 𝟐𝟐 = 𝟎. 𝟓𝟔
𝛍 = 𝟑. 𝟒𝟏 𝝈𝟐
= 𝟏. 𝟓𝟗

20. 1.What is the mean?
𝝁 = ෍[𝑿 ∗ 𝑷(𝑿)]
= 𝟎 + 𝟎. 𝟎𝟓 + 𝟎. 𝟐𝟒 + 𝟎. 𝟗𝟎 + 𝟏. 𝟏𝟐 + 𝟏. 𝟏𝟎
𝝁 = 𝟑. 𝟒𝟏
SOLUTION:

21. 2. What is the standard deviation?
𝜎2
= ෍ (𝑋 − 𝜇)²∙ 𝑃(𝑋)
= 𝟎. 𝟑𝟓 + 𝟎. 𝟐𝟗 + 𝟎. 𝟐𝟒 + 𝟎. 𝟎𝟓 + 𝟎. 𝟏𝟎 + 𝟓𝟔
𝝈𝟐
= 𝟏. 𝟓𝟗
SOLUTION:

22. 3. What is the standard deviation?
𝜎 = σ (𝑋 − 𝜇)²∙ 𝑃(𝑋) 𝑂𝑅 𝜎 = 𝜎2
𝜎 = 1.59
𝝈 = 𝟏. 𝟐𝟔
SOLUTION:

23. 1. The number of shoes sold per day at a retail store is shown in the
table below. Illustrate the mean, variance, and standard deviation
of this distribution.
Write all the necessary formula and show the complete solution.
Formula to be used: (a)Mean, (b)Variance,(c) Standard
Deviation
X 19 20 21 22 23
P(X) 0.4 0.2 0.2 0.1 0.1
ACTIVITY 1

24. 1. The Land Bank of the Philippines Manager claimed
that each saving account customer has several
credit cards. The following distribution showing the
number of credits cards people own.
a. Calculate the mean, variance, and standard deviation
of a discrete random variable.
X 0 1 2 3 4
P(X) 0.18 0.44 0.27 0.08 0.03
ASSIGNMENT

25. 2. Suppose an unfair die is rolled and let X be the
random variable representing the number of dots
that would appear with a probability distribution
below.
a. Calculate the mean, variance, and standard deviation
of a discrete random variable.
OUTCOME (X) 1 2 3 4 5 6
Probability 𝑃(X) 0.1 0.1 0.1 0.5 0.1 0.1

26. 1. The number of cellular phones sold per day at
the E-Cell Retail Store with the corresponding
probabilities is shown in the table below.
Compute the mean, variance, and standard
deviation and interpret the result.
(𝑥) 15 18 19 20 22
Probability 𝑃(𝑥) 0.30 0.20 0.20 0.15 0.15