Exploring Random Variables

1. EXPLORING RANDOM VARIABLES

2. Lesson Objectives
At the end of this lesson, you are expected to:
• illustrate a random variable;
• classify random variables as discrete or
continuous; and
• find the possible values of a random variable.

3. Pre-Assessment
Recap:
The set of all possible outcomes of an
experiment is called the sample space.

4. Lesson Introduction
If three coins are tossed, what numbers can be
assigned for the frequency of heads that will
occur?
If three cards are drawn from a deck, what
number can be assigned for the frequency of face
cards that will occur?
The answers to these questions require an
understanding of random variables.

5. Discussion Points
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 shall assign a
value.

6. Discussion Points
Suppose three cell phones are tested at random.
We want to find out the number of defective cell
phones that occur.
Possible Outcomes
NNN NDD
NND DND
NDN DDN
DNN DDD

7. Discussion Points
To each outcome in the sample space we shall assign a
value.
0 - If there is no defective cell phone
1- if there is 1 defective cell phone
2- if there are two defective cell phones
3 -if there are three defective cell phones
The number of defective cell phones is a random
variable.

8. Discussion Points
The possible values of this random variable are
0, 1, 2, and 3.

9. Discussion Points
A random variable is a function that associates
a real number to each element in the sample
space. It is a variable whose values are
determined by chance.

10. Example 1
Tossing Three Coins
Suppose three coins are tossed. Let Y be the
random variable representing the number of
tails that occur. Find the values of the random
variable Y. Complete the table below.

11. Solution to Example 1
The possible values of the random variable Y are
0, 1, 2, and 3.

12. Example 2
Drawing Balls from an Urn
Two balls 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. Find the
values of the random variable Z. Complete the
table below.

13. Solution to Example 2
The possible values of the random variable Z are
0, 1, and 2.

14. Discussion Points
A random variable is a discrete random variable if
its set of possible outcomes is countable. Mostly,
discrete random variables represent count data,
such as the number of defective chairs produced in
a factory.
For Example 1, the possible values of random
variable Y are 0, 1, 2, and 3. The possible values for
random variable Z in Example 2, are 0, 1, and 2.
Random variables Y and Z are discrete random
variables.

15. Discussion Points
A random variable is a continuous random
variable if it takes on values on a continuous
scale. Often, continuous random variables
represent measured data, such as heights,
weights, and temperatures.

16. Discussion Points
Example of Continuous Random Variable
Suppose an experiment is conducted to
determine the distance that a certain type of car
will travel using 10 liters of gasoline over a
prescribed test course. If distance is a random
variable, then we have an infinite number of
distances that cannot be equated to the number
of whole numbers. This is an example of a
continuous random variable.

17. Exercise 1
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.

18. Exercise 2
A shipment of five computers contains two that
are slightly defective. If a retailer receives three of
these computers at random, list the elements of
the sample space S using the letters D and N for
defective and non-defective computers,
respectively. To each sample point assign a value x
of the random variable X representing the number
of computers purchased by the retailer which are
slightly defective.

19. Exercise 3
Let T be a random variable giving the number
of heads plus the number of tails in three
tosses of a coin. List the elements of the
sample space S for the three tosses of the coin
and assign a value to each sample point.

20. Exercise 4
Classify the following random variables as discrete or
continuous.
a) the number of defective computers produced by
a manufacturer
b) the weight of newborns each year in a hospital
c) the number of siblings in a family of a region
d) the amount of paint utilized in a building project
e) the number of dropout in a school district for a
period of 10 years

21. Summary
A random variable is a function that associates
a real number to each element in the sample
space. It is a variable whose values are
determined by chance.

22. Summary
• A random variable is a discrete random variable if its
set of possible outcomes is countable. Mostly,
discrete random variables represent count data, such
as the number of defective chairs produced in a
factory.
• A random variable is a continuous random variable if
it takes on values on a continuous scale. Often,
continuous random variables represent measured
data, such as heights, weights, and temperatures.