1. The document discusses probability mass functions of discrete random variables, including their properties and examples of constructing probability mass functions in tables and histograms. 2. It provides examples of calculating probabilities of discrete random variables like the number of tails in coin flips and the sums of coin flips and die rolls. 3. The document also covers histograms as a way to graphically represent probability mass functions of discrete random variables.

1. STATISTICS AND
PROBABILITY

2. A B C D E F G H I J K
1
2
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At the end of this lesson, the learner should be able to:
• correctly construct a probability mass function of a given discrete
random variable;
• accurately graph values and probabilities of a discrete random variable
in a histogram; and
• accurately illustrate discrete random variables in real-life situations.
Objectives – Unit 2 Lesson 1: Probability Mass Function of a
Discrete Random Variable

3. What is Probability Mass
Function Discrete Random
Variable?

4. Properties of Probability Mass Function of a Discrete Random Variable
1. 𝑓(𝑥) = 𝑃(𝑋 = 𝑥)
2. 0 ≤ 𝑓(𝑥) ≤ 1
3. The sum of the probabilities, 𝑓 𝑥𝑖 , is equal to 1.
Probability Mass Function of a Discrete Random
Variable
PMF – is a function which describes the probability associated with the random
variable 𝑋. This function is named 𝑃 𝑋 𝑜𝑟𝑃 𝑋 = 𝑥 to avoid confusion. 𝑃 𝑋 =
𝑥 corresponds to the probability that the random variable 𝑋 take the value 𝑥.

5. Example 1: A coin is flipped three times. Let 𝑋 represent the number of tails that
appear in flipping a coin.
Probability Mass Function of a Discrete Random Variable
C
O
I
N
Head
Head Head
Tale
Tale Head
Tale
Tale
Head Head
Tale
Tale Head
Tale
HHH 0
HHT 1
HTH 1
HTT 2
THH 1
THT 2
TTH 2
TTT 3
X = 0, 1, 2, 3

6. 𝑋 0 1 2 3
𝑃(𝑋 = 𝑥) 1
8
3
8
3
8
1
8
Example 1.1: A coin is flipped three times. Let 𝑋 represent the number of tails that
appear in flipping a coin. Create a table to represent the probability mass
function of this event.
Probability Mass Function of a Discrete Random Variable
HHH 0
HHT 1
HTH 1
HTT 2
THH 1
THT 2
TTH 2
TTT 3

7. 𝑋 0 1 2 3
𝑃(𝑋 = 𝑥) 1
8
3
8
3
8
1
8
Example 1.2: A coin is flipped three times. Let 𝑋 represent the number of tails that
appear in flipping a coin. Find the probability mass function of
𝑋.
Probability Mass Function of a Discrete Random Variable
Therefore, 𝑃 𝑋 = 0 =
1
8
, 𝑃 𝑋 = 1 =
3
8
, 𝑃 𝑋 = 2 =
3
8
and 𝑃 𝑋 = 3 =
1
8
.

8. Example 2: A coin and a die are thrown once, at the same time given that for the
coin the head is represented by number 1 while the tail is represented by
number 2. If R is the random variable that represents the sum of the
results, what are the possible values of R?
Probability Mass Function of a Discrete Random Variable
1 1
1 2
1 3
1 4
1 5
1 6
2
3
4
5
6
7
2 1
2 2
2 3
2 4
2 5
2 6
3
4
5
6
7
8
R= 2, 3, 4, 5,
6, 7 & 8

9. Example 2.1: A coin and a die are thrown once, at the same time given that for the
coin the head is represented by number 1 while the tail is represented
by number 2. If R is the random variable that represents the sum of the
results, create a table to represent the probability mass function of this
event.
Probability Mass Function of a Discrete Random Variable
1 1
1 2
1 3
1 4
1 5
1 6
2
3
4
5
6
7
2 1
2 2
2 3
2 4
2 5
2 6
3
4
5
6
7
8
R = 2, 3, 4, 5, 6, 7 & 8
𝑹 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖
𝑷(𝑹 = 𝒓) 𝟏
𝟏𝟐
𝟐
𝟏𝟐
𝟐
𝟏𝟐
𝟐
𝟏𝟐
𝟐
𝟏𝟐
𝟐
𝟏𝟐
𝟏
𝟏𝟐

10. Example 2.2: A coin and a die are thrown once, at the same time given that for the
coin the head is represented by number 1 while the tail is represented
by number 2. If R is the random variable that represents the sum of the
results, find the probability mass function of R.
Probability Mass Function of a Discrete Random Variable
𝑹 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖
𝑷(𝑹 = 𝒓) 𝟏
𝟏𝟐
𝟐
𝟏𝟐
𝟐
𝟏𝟐
𝟐
𝟏𝟐
𝟐
𝟏𝟐
𝟐
𝟏𝟐
𝟏
𝟏𝟐
Therefore, 𝑃 𝑅 = 2 =
1
12
, 𝑃 𝑅 = 3 =
2
12
, 𝑃 𝑅 = 4 =
2
12
, 𝑃 𝑅 = 5 =
2
12
,
𝑃 𝑅 = 6 =
2
12
, 𝑃 𝑅 = 7 =
2
12
and 𝑃 𝑅 = 8 =
1
12
.

11. What is Histogram?

12. The possible values of the
discrete random variable are
on the horizontal axis while
its probabilities are on the
vertical axis. The total area
under a histogram is 1.
HISTOGRAM - a graph of a probability mass function. This is a graph that
displays the data by using vertical bars of various heights to
represent the probability of a certain random variable.
X
P(X)

13. 𝑋 0 1 2 3
𝑃(𝑋 = 𝑥) 1
8
3
8
3
8
1
8
Example 3: A coin is flipped three times. Let 𝑋 represent the number of tails that
appear in flipping a coin. Find the probability mass function using
histogram.
Probability Mass Function of a Discrete Random Variable

14. Example 4: A coin and a die are thrown once, at the same time given that for the
coin the head is represented by number 1 while the tail is represented by
number 2. If R is the random variable that represents the sum of the
results, Find the probability mass function using histogram.
Probability Mass Function of a Discrete Random Variable
𝑹 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖
𝑷(𝑹 = 𝒓) 𝟏
𝟏𝟐
𝟐
𝟏𝟐
𝟐
𝟏𝟐
𝟐
𝟏𝟐
𝟐
𝟏𝟐
𝟐
𝟏𝟐
𝟏
𝟏𝟐
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
2 3 4 5 6 7 8
R

15. Example 5: Two coins are tossed. Let 𝑆 be the number of heads that occur. Construct the
probability distribution for the random variable 𝑆 and its corresponding histogram.
HH 2
HT 1
TH 1
TT 0
S = 0, 1, 2
C
O
I
N
Head
Head
Tale
Tale
Head
Tale 𝑺 𝟎 𝟏 𝟐
𝑷(𝑺 = 𝒔) 𝟏
𝟒
𝟐
𝟒
𝟏
𝟒

16. 𝑆 0 1 2
𝑃(𝑆 = 𝑠) 1
4
2
4
1
4
P(S)
S

17. Example 6: A certain university has a lab with six computers reserved for students specializing in a
certain subject. Assume that X represents the number of these computers that are in use at a
particular time of a day. The probability distribution of X is given below. Find the probability that at
least 3 computers are in use at a particular time of the day.
𝑋 0 1 2 3 4 5 6
𝑃(𝑋 = 𝑥) 0.10 0.10 0.15 0.25 0.25 0.10 0.05
Solution:
𝑃 𝑋 ≥ 3 = 𝑃 𝑋 = 3 + 𝑃 𝑋 = 4 + 𝑃 𝑋 = 5 + 𝑃(𝑋 = 6)
𝑃 𝑋 ≥ 3 = 0.25 + 0.25 + 0.10 + 0.05
𝑷 𝑿 ≥ 𝟑 = 𝟎. 𝟔𝟓
Thus, there is 65% chance are at least 3
computers are in use if we visit it in a particular
time of day.

21. TRY IT!
Toss a pair of dice. Winnings are equal to the difference of the
numbers on the two dice. Let M denote the winnings after
playing the game once.
a) Find the possible values of M.
b) Find the probability mass function of M.
c) Construct its corresponding histogram.

22. 1 1
1 2
1 3
1 4
1 5
1 6
4 1
4 2
4 3
4 4
4 5
4 6
2 1
2 2
2 3
2 4
2 5
2 6
5 1
5 2
5 3
5 4
5 5
5 6
3 1
3 2
3 3
3 4
3 5
3 6
6 1
6 2
6 3
6 4
6 5
6 6
Thus, the possible
values of M are
0, -1, -2, -3, -4, -5,
1, 2, 3, 4 & 5.
0
-1
-2
-3
-4
-5
3
2
1
0
-1
-2
1
0
-1
-2
-3
-4
4
3
2
1
0
-1
2
1
0
-1
-2
-3
5
4
3
2
1
0
Toss a pair of dice.
Winnings are equal to
the difference of the
numbers on the two
dice. Let M denote the
winnings after playing
the game once.
a) Find the possible
values of M.

23. Toss a pair of dice. Winnings are equal to the difference of the numbers on the
two dice. Let M denote the winnings after playing the game once.
b) Find the probability mass
function of M. Thus, the possible values of M are
0, -1, -2, -3, -4, -5, 1, 2, 3, 4 & 5.
-5 1
-4 2
-3 3
-2 4
2 4
3 3
4 2
5 1
𝑴 -5 -4 -3 -2 -1 0 1 2 3 4 5
𝑷(𝑴) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
-1 5
0 6
1 5

24. 𝑃 𝑀 = −4 =
2
36
𝑃 𝑀 = −5 =
1
36
𝑃 𝑀 = −2 =
4
36
𝑃 𝑀 = −3 =
3
36
𝑃 𝑀 = 0 =
6
36
𝑃 𝑀 = −1 =
5
36
𝑃 𝑀 = 2 =
4
36
𝑃 𝑀 = 1 =
5
36
𝑃 𝑀 = 3 =
3
36
𝑃 𝑀 = 5 =
1
36
𝑃 𝑀 = 4 =
2
36

25. A B C D E F G H I J K
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Check your To Do’s on your Quipper Account, answer the
assignment entitled Probability Mass Function of a Discrete
Random Variable.
Online Quiz: