Distributions of Random Variables

SQQS1013 Elementary Statistics
DISTRIBUTION OFDISTRIBUTION OF
RANDOMRANDOM
VARIABLESVARIABLES
4.1 RANDOM VARIABLE
Definition:
A random variable is a variable whose value is determined by the
outcome of a random experiment
• Supposed one family is randomly selected from the population. The process
of random selection is called random or chance experiment.
• Let X be the number of vehicles owned by the selected family (0, 1, 2, …, n).
Therefore the first column represents five possible values (0, 1, 2, 3 and 4) of
vehicles owned by the selected family.
• This table shows that 30 families do not have vehicle, 470 families have
1 vehicle, 850 families have 2 vehicles, 490 families have 3 vehicles and
160 families have 4 vehicles.
• In general, a random variable is denoted by X or Y.
4.2 DISCRETE RANDOM VARIABLE
Chapter 4: Distribution of Random Variables 1

Definition: A random variable that assumes countable values is called
discrete random variable.
• Number of houses sold by a developer in a given month.
• Number of cars rented at a rental shop during a given month.
• Number of report received at the police station on a given day.
• Number of fish caught on a fishing trip.
4.2 PROBABILITY DISTRIBUTION OF A DISCRETE RANDOM
VARIABLE
Definition:
The probability distribution of a discrete random variable lists
all the possible values that the random variable can assume and
their corresponding probabilities.
• It is used to represent populations.
• The probability distribution can be presented in the form of a
mathematical formula, a table or a graph.
Consider the table below. Let X be the number of vehicles owned by a randomly
selected family. Write the probability distribution of X and graph for the data.
Example of Discrete Random Variables
Example 1

Solution:
X 0 1 2 3 4
P(x) 0.015 0.235 0.425 0.245 0.080
During the summer months, a rental agency keeps track of the number of cars it rents
each day during a period of 90 days and X denotes the number of cars rented per
day. Construct a probability distribution and graph for the data.
X Number of days
0 45
1 30
2 15
Total 90
Solution:
When
Hence, the probability distribution for X:
X 0 1 2
P(x) 0.5 0.33 0.17
Whereas the graph is shown below:
P(x)
0.05
0.10
0.15
0.25
0.20
0 1 2
X
3
0.30
0.35
0.40
0.45
4
Example 2

One small farm has 10 cows where 6 of them are male and the rest are female. A
veterinary in country XY wants to study on the foot and mouth disease that attacks the
cows. Therefore, she randomly selects without replacement two cows as a sample
from the farm. Based on the study, construct a probability distribution which X is the
random sample representing the number of male cows that being selected as a
sample (Use tree diagram to illustrate the above event).
• Conditions for probabilities for discrete random variable.
i) The probability assigned to each value of a random variable x must be
between 0 and 1.
0≤ P(x) ≤1, for each value of x
Example 3

ii) The sum of the probabilities assigned to all possible values of x is equal
to 1.
∑P(x) = 1
The following table lists the probability distribution of car sales per day in a used car
shop based on passed data.
Car Sales per day, X 0 1 2 3
P(x) 0.10 0.25 0.30 0.35
Find the probability that the number of car sales per day is,
a) none
b) exactly 1
c) 1 to 3
d) more than 1
e) at most 2
Example 4

4.3.1 Mean of a Discrete Random Variables
Definition:
The mean of a discrete random variable X is the value that is
expected to occur repetition, on average, if an experiment is
repeated a large number of times.
• It is denoted by µ and calculated as:
∑= )(. xPXµ
• The mean of a discrete random variable X is also called as its expected
value and is denoted by E(X),
E(X) =∑ )(. xPX
4.3.2 Standard Deviation of a Discrete Random Variable
Definition:
The standard deviation of a discrete random variable X measures
the spread of its probability distribution.
• It is denoted by σ and calculated as:
2 2
( )x P xσ µ= −∑
FORMULA
ξ∆Σ λϖ
β
FORMULA
ξ∆Σ λϖ
β
FORMULA
ξ∆Σ λϖ
β

• A higher value for the standard deviation of a discrete random variable
indicates that X can assume value over a large range about the mean.
• In contrast, a smaller value for the standard deviation indicates the most of
the value that X can assume clustered closely about the mean.
The following table lists the probability distribution of car sales per day in a used car
dealer based on passed data. P(x) is the probability of the corresponding value of X =
x. Calculate the expected number of sales per day and followed by standard
deviation.
X P(x)
0 0.1
1 0.25
2 0.3
3 0.35
Total 1.00
Solution:
During the summer months, a rental agency keeps track of the number of chain
saws it rents each day during a period of 90 days and X denotes the number of
saws rented per day. What is the expected number of saws rented per day?
Then, find the standard deviation.
X 0 1 2
P(x) 0.5 0.33 0.17
Example 5
Example 6

Solution:
Mean
Standard Deviation
4.4 CUMULATIVE DISTRIBUTION FUNCTION
Definition:
The cumulative distribution function (CDF) for a random variable
X is a rule or table that provides the probabilities ( )P X x≤ for any
real number x.
• Generally the term cumulative probability refers to the probabilities that X less
than or greater than or equal to a particular value.
• For a discrete random variable, the cumulative probability ( )P X x≤ is a
function ( )F x ,
Where;
FORMULA
ξ∆Σ λϖ
β

0
( ) ( ) ( )
t
x x
F x P X x P X x
=
= ≤ = =∑
and
( )P X x= ,
Where; 0 1 2, , ...x x x x= is the probability distribution function for X.
A discrete random variable X has the following probability distribution.
Construct the cumulative distribution of X.
Solution:
Chapter 4: Distribution of Random Variables
X 0 1 2 3
( )P X x=
1
30
3
10
1
2
1
6
X 0 1 2 3
P(x)
F(x)
9
Example 7
FORMULA
ξ∆Σ λϖ
β
Example 8

A discrete random variable X has the following cumulative distribution.
1
, for 0 1
21
3
, for 1 2
21
6
, for 2 3
( ) 21
10
, for 3 4
21
15
, for 4 5
21
1 , for 5
x
x
x
F x
x
x
x

≤ <

 ≤ <


 ≤ <
= 

≤ <


≤ <

≥

a) Construct the probability distribution of X.
X 0 1 2 3 4 5
P(x)
F(x)
b) Construct the graph of the:
i. probability distribution of X
.
ii. cumulative distribution of X

During the school holiday, the manager of Victory Hotel records the number of
room bookings being cancelled each day during a period of 50 days, the results
are shown below and Y denotes the number of room bookings being cancelled
per day.
Number of room bookings being cancelled per day, Y Number of days
0 2
1 4
2 7
3 8
4 13
5 10
6 3
7 3
a) Construct the probability distribution of X.
Y
P(y
)
b) Then, draw a bar chart for the probability distribution.
Example 9 (Overall
Example)
12

c) The manager expects that five room bookings were cancelled for a day. Is
the manager expectation true? Explain.
d) Find the probability that at most three room bookings were cancelled.
e) Find the standard deviation for the number of room bookings being
cancelled.
Y
P(y)
Y2
.P(y)

4.5 CONTINUOUS RANDOM VARIABLE
Definition:
A random variable that can assume any value contained in one or
more intervals is called a continuous random variable.
• Examples of continuous random variables,
 The weight of a person.
 The time taken to complete a 100 meter dash.
 The duration of a battery.
 The height of a building.

EXERCISE
1. The following table gives the probability distribution of a discrete random
variable X.
Find the following probability:
a) exactly 1.
b) at most 1.
c) at least 3.
d) 2 to 5.
e) more than 3.
2. The following table lists the frequency distribution of the data collected by a
local research agency.
a) Construct the probability distribution table.
b) Let X denote the number of television sets owned by a randomly
selected family from this town. Find the following probabilities:
i. exactly 3.
ii. more than 2.
iii. at most 2.
iv. 1 to 3.
v. at least 4.
3. According to a survey 65% university students smokes. Three students are
randomly selected from this university. Let X denote the number of students
in this sample who does not smokes.
a) Draw a tree diagram for this problem.
b) Construct the probability distribution table.
c) Let X denote the number of students who does not smoking is
selected randomly. Find the following probability:
x 0 1 2 3 4 5
P(x) 0.03 0.17 0.22 0.31 0.15 0.12
Number of TV sets
own
0 1 2 3 4 5 6
Number of families 110 891 329 340 151 76 103
15

i. at most 1.
ii. 1 to 2.
iii. at least 2.
iv. more than 1.
4. The following table gives the probability distribution of the number of
camcorders sold on a given day at an electronic store.
Calculate the mean and standard deviation for this probability distribution.
5. According to a survey, 30% of adults are against using animals for research.
Assume that this result holds true for the current population of all adults. Let
x be the number of adults who agrees using animals for research in a
random sample of three adults. Obtain:
a) the probability distribution of X.
b) mean.
c) standard deviation
6. In a genetics investigation, cat litters with ten kittens are studied which of
three are male. The scientist selects three kittens randomly. Let X as the
number of female kittens that being selected and construct probability
distribution for X (you may use tree diagram to represent the above event).
Based on the probability distribution obtained, find the:
a) mean.
b) standard deviation.
7. A box holds 5 whites and 3 black marbles. If two marbles are drawn
randomly without replacement and X denoted the number of white marbles,
a) Find the probability distribution of X.
b) Plot the cumulative frequency distribution (CFD) of X.
8. The following table is the probability distribution for the number of traffic
accidents occur daily in a small city.
Number of
accidents (Y)
0 1 2 3 4 5
P(y) 0.10 0.20 9a 3a a a
Camcorder sold 0 1 2 3 4 5 6
Probability 0.05 0.12 0.19 0.30 0.18 0.10 0.06
16

a) Find the probability of:
i. exactly three accidents occur daily.
ii. between one and four accidents occur daily.
iii. at least three accidents occur daily.
iv. more than five accidents occur daily and explain your answer.
b) Traffic Department of that small city expects that 5 accidents occur
daily. Do you agree? Justify your opinion.
c) Compute the standard deviation.
9. The manager of large computer network has developed the following
probability distribution of the number of interruptions per day:
Interruptions(Y) 0 1 2 3 4 5 6
P(y) 0.32 0.35 0.18 0.08 0.04 0.02 0.01
a) Find the probability of:
i. more than three interruptions per day.
ii. from one to five interruptions per day.
iii. at least an interruption per day.
b) Compute the expected value.
c) Compute the standard deviation.
10. You are trying to develop a strategy for investing in two different stocks. The
anticipated annual return for a RM1,000 investment in each stock has the
following probability distribution.
Returns (RM), X
P(x)Stock A Stock B
-100 50 0.1
0 150 0.3
80 -20 0.3
150 -100 a
a) Find the value of a.
b) Compute,
i. expected return for Stock A and Stock B.
ii. standard deviation for both stocks.
c) Would you invest in Stock A or Stock B? Explain.
11. Classify each of the following random variables as discrete or continuous.
a) The time left on a parking meter.
b) The number of goals scored by a football player.
c) The total pounds of fish caught on a fishing trip.
d) The number of cans in a vending machine.

e) The time spent by a doctor examining a patient.
f) The amount of petrol filled in the car.
g) The price of a concert ticket.

Matrix No:________________ Group: _________
TUTORIAL CHAPTER 4
The random variable X represents the number of children per family in a rural;
area in Ohio, with the probability distribution: p(x) = 0.05x, x = 2, 3, 4, 5, or 6.
1. Express the probability distribution in tabular form.
ANSWER:
x 2 3 4 5 6
p(x)
2. Find the expected number of children per family.
3. Find the variance and standard deviation of X.
Tutorial
19

4. Find the following probabilities:
a. P(X ≥4)
b. P(X > 4)
c. P(3 ≤X ≤5)
d. P(2 < X < 4)
e. P(X = 4.5)
Tutorial
20

Distributions of Random Variables

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