Understanding Random Variables

What is a Random Variable?

Random Variable

• It is also called a stochastic variable.
• It is a numerical description of the outcome of a statistical experiment.
• It is a set of possible values from a random experiment.
• It is a variable, denoted as or any capital letter of the alphabet.

There are two types of random variables:

• Discrete
  • It is an outcome obtained via a counting process.
  • A random variable is discrete when the set of possible outcomes are countable.
  • Examples: number of soda cans, number of chairs, number of students in class
• Continuous
  • It is an outcome obtained from a measurement.
  • A random variable is discrete when the set of possible outcomes takes on values on a continuous scale.
  • Examples: height, weight, volume of water, amount of solution in an alcohol, etc.

Problems involving Random Variables

Problem

Suppose two coins are tossed. Let be the random variable representing the number of heads that occur. Find the values of the random variable .

Solution

Step 1: First, create a tree diagram to find all possible outcomes for tossing two coins.

stateDiagram-v2
	t1a : H
	t1b : T
	t2aa : H
	t2ab : T
	t2ba : H
	t2bb : T
	
	direction LR
	t1a --> t2aa
	t1a --> t2ab
	t1b --> t2ba
	t1b --> t2bb

Step 2: Next, build the sample space.

stateDiagram-v2
	t1a : H
	t1b : T
	t2aa : H
	t2ab : T
	t2ba : H
	t2bb : T
	s1 : HH
	s2 : HT
	s3 : TH
	s4 : TT 
	
	direction LR
	t1a --> t2aa 
	t1a --> t2ab
	t1b --> t2ba
	t1b --> t2bb
	
	t2aa --> s1
	t2ab --> s2
	t2ba --> s3
	t2bb --> s4

Therefore, the sample space is:

Step 3: Since the random variable is the number of heads that might occur, we count the number of heads inside the sample space.

Possible Outcomes	Number of Heads

Therefore, the possible values of the random variable are , , and .

Problem

Suppose three coins are tossed. Let be the random variable representing the number of tails that occur. Find the values of the random variable .

Solution

Step 1: Create a tree diagram that represents all possible outcomes of the random variable, then determine the sample space.

stateDiagram-v2
	t1a : H
	t1b : T
	t2aa : H
	t2ab : T
	t2ba : H
	t2bb : T
	t3aaa : H
	t3aab:  T
	t3aba : H
	t3abb : T
	t3baa: H
	t3bab : T
	t3bba : H
	t3bbb : T
	s1 : HHH
	s2:  HHT
	s3 : HTH
	s4 : HTT
	s5: THH
	s6: THT
	s7: TTH
	s8 : TTT
	
	direction LR
	t1a --> t2aa
	t1a --> t2ab
	t1b --> t2ba 
	t1b --> t2bb
	
	t2aa --> t3aaa
	t2aa --> t3aab
	t2ab --> t3aba
	t2ab --> t3abb
	t2ba --> t3baa
	t2ba --> t3bab
	t2bb --> t3bba
	t2bb --> t3bbb
	
	t3aaa --> s1
	t3aab --> s2
	t3aba --> s3
	t3abb --> s4
	t3baa --> s5
	t3bab --> s6
	t3bba --> s7
	t3bbb --> s8

Therefore, the set of all sample space outcomes are:

Step 2: Since the random variable is the number of tails, we then count the number of tails for each outcome.

Outcome	Number of Tails

Therefore the possible values for the random variable are:

Problem

Two balls are drawn in succession without replacement from a box containing 5 red balls and 6 blue balls. Let be the random variable representing the number of blue balls. Find the values of the random variable .

Solution

Step 1: We build a tree diagram to find the number of possible outcomes of red and blue balls in two draws.

stateDiagram-v2
	t1a : R
	t1b : B
	t2aa : R
	t2ab : B
	t2ba : R
	t2bb : B
	s1 : RR
	s2 : RB
	s3 : BR
	s4 : BB 
	
	direction LR
	t1a --> t2aa 
	t1a --> t2ab
	t1b --> t2ba
	t1b --> t2bb
	
	t2aa --> s1
	t2ab --> s2
	t2ba --> s3
	t2bb --> s4

Therefore, the sample space is:

Step 2: Since our random variable is the number of blue balls drawn, we then count the number of blue balls for each outcome.

Possible Outcomes	Number of Blue Balls

Therefore, the possible values of the random variable are , , and .

Problem

Write the possible values of each random variable.

• : The number of even number outcomes in a roll of a siz-sided die.

Solution

Since the even numbers in a six-sided die are only , , and , therefore

• : Weight (in mg) of a powder that does not exceed 80 mg.

Solution

Since weight cannot be negative, and it cannot be greater than , therefore

• : Scores of a student in a -item test

Solution

A student cannot score higher than since it is a -item test. Likewise, a student cannot also score less than as well. Therefore,

• : A product of two numbers taken from two boxes containing numbers from 0 to 5.

  Solution

  To find the values of , it’s better that we use a table to demonstrate this:

  We assign the rows to be the number that you take from the first box, and the columns to be the number that you take from the second box

  Multiplying the rows and the columns should give you each possible outcome.

  Therefore, the values of are:

What is a Discrete Probability Distribution?

• It is a table that gives a list of probability values, along with their associated value in the range of a discrete random variable.
• It is also known as the probability mass function.

Properties of a Probability Distribution

• Each probability values ranges from 0 to 1
• The sum of all individual probabilities in the distribution is equal to .

Examples involving Discrete Probability Distributions

Problem

Three coins are tossed, and the random variable gives the number of heads.
Create a probability distribution using the random variable .

Solution

Step 1: Determine the range space.
The range space is all possible values of the random variable . Since tossing three coins gives us the sample space:

Therefore the values of the random variable or the number of possible heads are .

Step 2: Create a Probability Table
Our next goal is to find the frequency.
To find the frequency, we ask ourselves how many times does an outcome has heads? How about head? or heads?

• Since only outcome only has 0 heads, therefore the frequency is .
• Since outcomes have 1 head, the frequency is
• Since outcomes have 2 heads, the frequency is as well.
• Since only outcome have 3 heads, the frequency is simply .

Therefore, we write this as:

Number of Heads
Frequency

Step 3: Computing the Probability
To compute the probability of an event, we use the formula:

Since there are total outcomes, we simply divide the frequency by .

Number of Heads
Frequency

Problem

An investor has 5 stocks that she follows each day. The random variable being studied is . Based on the table below, find

Solution

Since we know that the sum of all probabilities are equal to , therefore:

We can substitute all known values first, then we solve for .

Therefore, the probability that the investor has zero stocks is .

Problem

A random variable can take the values , , and . If:

Are the given values a valid probability distribution? Why or why not?

Solution

In a valid probability distribution, all probabilities must add up to .
Therefore:

Since the probabilities does not add up to , then it is not a valid probability distribution.