Module 1 Understanding Random Variables
Introduction
Recall that an experiment is an activity that produces outcomes.
Some examples of which are
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tossing three coins and counting the number of heads,
recording the time a person can squat before he/she gets exhausted,
counting the class attendance of students for today, etc.
In this lecture, we will be talking about a way to map the outcomes of these statistical experiments
determined by probabilities to a number.
Definition of a Random Variable
A random variable is a numerical quantity that represents the outcome of an experiment. It is usually
denoted by an uppercase letter of the English alphabet.
Some examples of random variable:
Q = the number of times the coin turns up a head in five trials of tossing it
R = the sum when you rolled a pair of dice S = the total running time of a track and field athlete
T= the number of points of an MVP player A random variable is a function that maps the outcomes of a
random process to a numeric value. That is,
X: Outcome= Number
Two Types Random Variables
1. Discrete Random Variables
2. Continuous Random Variables
Two Types Random Variables
A discrete random variable is a random variable associated with numerical values or
set of possible outcomes that are countable or whole numbers only
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Some Examples of Discrete Random Variables:
Number of defective iPads produced by Apple in a specific plant
Number of female athletes in a University
Number of patient arrivals per hour at a certain hospital
The number of questions answered correctly in specific exam.
The number of languages a certain group of individuals speak.
A continuous random variable is a random variable where the observed numerical
values are in a continuous scale. This can take on non-integers as this can take on any
of the numbers contained within the number line.
Some Examples of Continuous Random Variables:
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Amount of sugar in a cup of milk tea
Speed of vehicles travelling along NLEX
Time needed to finish an examination
Height of basketball players
Diameter of a table
How can you determine the possible values of a random variable?
X: Outcome→ Number
Determining the Possible Outcomes and Value of a Random Variable Identifying the possible outcomes
(or otherwise known as writing the sample space of the random variable) and the value of the random
variable.
Remember that,
1. A collection of all possible outcomes which is usually denoted by the letter "S" are called sample
space, and n(S) for the number of possible outcomes.
2. A collection of possible sample points of an event is denoted by the letter "E" are called sample points,
and n(E) for the number of sample points of an event.
3. The number of sample points of an event is a subset of the possible outcomes of an experiment
Example 1
Suppose a coin is tossed twice. Let X be the number of heads that occur. Determine the possible values
of the random variable X.
Step 1: List the sample space of the experiment.
H-Head
T-Tail
S={HH, HT, TH, TT}
Outcomes of the Experiment
HH
HT
TH
TT
Number of heads (x)
2
1
1
0
Example 2
A box contains two different colors of balls red and blue. Three balls are picked one at a time then
returned in the box. Let Z be the random variable of red balls drawn from
the box.
Step 1: List the sample space of the experiment.
R - red ball
B - blue ball S = {RRR, RRB, RBR, RBB, BRR, BRB, BBR, BBB}
Example 2
Step 2: Count the possible number of red balls drawn from the box.
Outcomes of the Experiment
RRR
RRB
RBR
BRR
RBB
BRB
BBR
BBB
Number of red balls (x)
3
2
2
2
1
1
1
0
Supplementary Exercises
List all possible outcomes and find the values of the random variable:
1. A manufacturer produces laptops. Suppose three units are tested by the quality assurance team and
they want to find the number of defective units that occur. Let D represent the defective units and N the
non-defective units. Show the values of the random variable X representing the number of defective
units using a table.
2. In an experiment four coins are tossed. Let M be the random variable representing the number of tails
that occur. Find the values of the random variable M. Use a table.
Solution to supplementary exercise 1.
A manufacturer produces laptops. Suppose three units are tested by the quality assurance team and
they want to find the number of defective units that occur. Let D represent the defective units and N the
non-defective units.
Show the values of the random variable X representing the number of defective units using a table.