ILLUSTRATING A
RANDOM VARIABLE
Definition of basic terms:
◦Probability is a measure that is associated
with how certain we are of outcomes of a
particular experiment or activity.
◦An experiment is a planned operation
carried out under controlled conditions.
Definition of basic terms:
◦A result of an experiment is called an
outcome.
◦The sample space of an experiment is the
set of all possible outcomes.
◦A variable is a characteristic or attribute
that can assume different values. We use
capital letters to denote or represent a
variable.
Qualitative Variables:
◦Consist of categories or attributes which
have non-numerical characteristics.
◦express a categorical attribute
◦answer questions “what kind”
◦Examples: hair color, name of section,
gender, etc.
Quantitative Variables:
◦Consist of numbers representing counts or
measurements
◦have actual units of measure
◦answer questions such as “how much” or
“how many”
◦Examples: height, weight, area, age, etc.
DISCRETE VARIABLES
◦ Discrete variables can only assume specific values that
you cannot subdivide. Typically, you count them, and
the results are integers. For example, if you work at an
animal shelter, you’ll count the number of cats.
◦ Discrete data can only take on specific values. For
example, you might count 20 cats at the animal shelter.
These variables cannot have fractional or decimal
values. You can have 20 or 21 cats, but not 20.5
EXAMPLES
◦The number of books you check out from
the library.
◦The number of heads in a sequence of
coin tosses.
◦The result of rolling a die.
◦The number of patients in a hospital.
◦The population of a country.
Continuous variables
◦Continuous variables can assume any
numeric value and can be meaningfully
split into smaller parts. Consequently, they
have valid fractional and decimal values.
In fact, continuous data have an infinite
number of potential values between any
two points. Generally, you measure them
using a scale.
Continuous variables
◦Examples of continuous data
include weight, height, length,
time, and temperature.
Classify the following as Qualitative
(QL) or Quantitative (QT) variables.
1.Most preferred color
2.Number of Siblings
3.Weight (in kg)
4.Outcome of tossing a coin
5.Quality of breads in a bake shop
Example 2
Suppose two coins are tossed and we
are interested to determine the
number of tails that will come out. Let
us use T to represent the number of
tails that will come out. Determine the
values of the random variable T.
PE #1
Determine the values of the random
variables:
1. Three coins are tossed. Let H be the
number of heads that occur.
Determine the values of the random
variable H.
Example 2
Two balls are drawn in succession
without replacement from an urn
containing 5 orange balls and 6
violet balls. Let V be the random
variable representing the number of
violet balls. Find the values of the
random variable V.
Example 3
A basket contains 10 red balls and 4
white balls. If three balls are taken
from the basket one after the other,
determine the possible values of the
random variable R representing the
number of red balls.
DISCRETE AND
CONTINUOUS
VARIABLES
Q1_WEEK1_DAY 2
Two Types of Random Variable:
◦ Discrete Random Variable - a random variable
whose possible values form a finite or countable set
of numbers. Examples: the number of students in a
class, the year when a certain student was born,
etc.
◦ Continuous Random Variable – a random variable
that can assume infinite number of values in one or
more intervals. Examples: the amount of sugar in an
orange, the time required to run a mile, etc.
Discrete data key characteristics:
◦ You can count the data. It is usually units counted in whole
numbers.
◦ The values cannot be divided into smaller pieces and add
additional meaning.
◦ You cannot measure the data. By nature, discrete data cannot
be measured at all. For example, you can measure your weight
with the help of a scale. So, your weight is not a discrete data.
◦ It has a limited number of possible values e.g. days of the month.
◦ Discrete data is graphically displayed by a bar graph.
Examples of discrete data:
◦ The number of students in a class.
◦ The number of workers in a company.
◦ The number of parts damaged during transportation.
◦ Shoe sizes.
◦ Number of languages an individual speaks.
◦ The number of home runs in a baseball game.
◦ The number of test questions you answered correctly.
◦ Instruments in a shelf.
◦ The number of siblings a randomly selected individual has.
Continuous data key characteristics:
◦ In general, continuous variables are not counted.
◦ The values can be subdivided into smaller and smaller
pieces and they have additional meaning.
◦ The continuous data is measurable.
◦ It has an infinite number of possible values within an
interval.
◦ Continuous data is graphically displayed by histograms.
Examples of continuous data:
◦ The amount of time required to complete a project.
◦ The height of children.
◦ The amount of time it takes to sell shoes.
◦ The amount of rain, in inches, that falls in a storm.
◦ The square footage of a two-bedroom house.
◦ The weight of a truck.
◦ The speed of cars.
◦ Time to wake up.
FINDING THE RANDOM
VARIABLE
Q1_WEEK1_DAY 3
◦Suppose three cell phones are tested at random.
We want to find out the number of defective cell
phones that occur. Let D represent the defective
cell phone and N represent the non-defective
cell phone. If we let X be the random variable
representing the number of defective cell
phones, can you show the values of the random
variable X by completing the table below?
Steps in Evaluating Random
Variables
◦1. List the sample space of the
experiment.
◦2. Count the number of the random
variable in each outcome and assign
this number to this outcome.
◦3. Conclude you answer
FIND THE RANDOM VARIABLE
◦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.
Example 3: Ripe and Unripe Bananas
◦A basket contains 10 ripe and 4 unripe
bananas. If three bananas are taken
from the basket one after the other,
determine the possible values of the
random variable B representing the
number of ripe bananas.
Evaluation: Do what is asked. Show
your complete solution.
◦ 1. From a box containing 4 pink (P) balls and 3 violet (V)
balls, 3 balls are drawn in succession. Each ball is placed
back in the box before the next draw is made. Let Z be a
random variable representing the number of violet balls that
occur. Find the values of the random variable Z.
◦ 2. A basket contains 4 ripe and 3 unripe mangoes. If three
mangoes are taken from the basket one after the other,
determine the possible values of the random variable M
representing the number of unripe mangoes.
CONSTRUCTING
PROBABILITY
DISTRIBUTIONS
Q1_WEEK1_DAY 4
Try to discover:
◦Suppose three coins are tossed. Let
Y be the random variable
representing the number of tails
that occur. Find the probability of
each values of the random
variable Y.
Probability Distribution of a Discrete Random
Variable
is a correspondence that assigns
probabilities to the values of a
random variable. The probability
distribution of a discrete random
variable is also called the
probability mass function.
Properties of a Probability Distribution
1. The probability of each value of the
random variable must be between or
equal to 0 and 1. In symbol, we write it as
0 ≤ 𝑃(𝑋) ≤ 1.
2. The sum of the probabilities of all values of
the random variable must be equal to 1.
In symbol, we write it as ∑ 𝑃(𝑋) = 1.