BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon105 Learners

Last updated on July 9th, 2025

Math Whiteboard Illustration

Discrete Random Variable

Professor Greenline Explaining Math Concepts

In probability, a discrete random variable is a variable that can take a countable set of values, such as whole numbers or integers. These values represent the possible outcomes of a random experiment.

Discrete Random Variable for Indian Students
Professor Greenline from BrightChamps

What is a Discrete Random Variable?

There are two types of random variables: discrete and continuous. Discrete random variables take on a countable set of distinct values, such as 0, 1, 2, 3,  …. The outcomes of a random experiment are countable. The probability distribution of discrete random variables is represented using a probability mass function.

Professor Greenline from BrightChamps

Difference Between Discrete Random Variable and Continuous Random Variable

Random variables can be classified into two types: discrete and continuous. Here is the difference between them.

 

Discrete Random Variable Continuous Random Variable
  • A discrete random variable can take a finite or countably infinite set of values, mostly whole numbers.
  • A continuous random variable can take any value within a given range, resulting in an uncountably infinite number of possible values.
  • A probability mass function describes the probability of each possible value.
  • A probability density function represents how the probability is distributed over the range of possible values.
  • Example: The number of students in a classroom on a particular day
  • Example: The time it takes a person to solve a puzzle
  • A specific value can have a non-zero probability
  • For a continuous random variable, the probability of any exact value is zero. However, probabilities are assigned over intervals.
Professor Greenline from BrightChamps

What are Mean and Variance of Discrete Random Variables?

The mean and variance are used to describe the behavior of a discrete random variable. Here, we will discuss what the mean and variance of discrete random variables, denoted E[X], is its expected value.

 

Mean of Discrete Random Variable: 

The mean of a discrete random variable is the average value of a random variable. It is represented as E[X], where X is the random variable. The mean is also known as the expected value and weighted average. 

 

The formula to calculate the mean of a discrete random variable is:
E[X] = Σx xP (X = x)

Where P(X = x) is the probability mass function

 

Variance of a Discrete Random Variable: 

The average of the squared deviations of the random variable from its mean is the variance of the discrete random variable. It can be represented as Var[X] or σ2

 

The variance of discrete random variables is calculated using the formula: 
Var[X] = Σ(x - µ)2 P(X = x)

Here, µ is the mean, and p(X = x) is the probability of each value. 

Professor Greenline from BrightChamps

What are the Types of Discrete Random Variables?

The discrete random variable is a variable that can take on specific countable values. It represents the possible outcomes of a random experiment and the probability of each outcome. The types of discrete random variables are:   

 

  • Binomial Random Variables

 

  • Geometric Random Variables

 

  • Bernoulli Random Variables

 

  • Poisson Random Variables
Professor Greenline from BrightChamps

Binomial Random Variables

The binomial random variables are the possible outcomes of a binomial experiment. A binomial random variable involves a fixed number of independent Bernoulli trials, where each result is either a success or a failure. It is represented as X ∼ Bin(n, p), where X is the binomial random variable. 

 

For a binomial random variable, the probability mass function:
P (X = x) = (xn) px (1 - p)n - x

Professor Greenline from BrightChamps

Geometric Random Variables

The geometric random variables represent the number of trials needed to get the first success in a sequence of independent Bernoulli trials. In a geometric random variable, the probability of success is denoted by p, and the probability of failure is 1 - p

 

The geometric variable is represented by: X ∼ Geom(p)

For a geometric random variable, the probability mass function is: 
P(X = x) = (1 - p)x - 1 p

Professor Greenline from BrightChamps

Bernoulli Random Variables

A Bernoulli random variable represents an experiment with two possible outcomes, i.e., 1 for a success and 0 for a failure. It is represented by X ∼ Bern(p). 
The probability mass function of a Bernoulli random variable is: 
P(X = x) = { p           if  x = 01 -p     if  x = 1

Professor Greenline from BrightChamps

Poisson Random Variables

A Poisson random variable represents the number of events that occur within a fixed time interval of time or space, where the events occur independently and at a constant rate. It is denoted by X ∼ λ, where λ is the parameter of a Poisson distribution, which is always greater than 0. 

 

The probability mass function is: P(X = x) = λxe/x!, x = 0, 1, 2, … .

Professor Greenline from BrightChamps

What is the Probability Distribution of Discrete Random Variables?

The probability distribution of a discrete random variable is the probability of each of its possible outcomes. When constructing a distribution table, the probability should follow these conditions:

 

  • Each probability value must lie between 0 and 1; that is, 0 ≤ p(X = x) ≤ 1 for each possible outcome.

 

  • The sum of the probabilities of all possible outcomes must be equal to 1
Professor Greenline from BrightChamps

Real-world Applications of Discrete Random Variables

In fields like quality control, statistics, and finance, we use discrete random variables. In this section, we will learn some real-world applications of discrete random variables.

 

  • In quality control, we use discrete random variables to find the number of defects in a batch and also to identify the quality issues. 

 

  • To predict the number of incidents that occur within a given period, we use discrete random variables. For example, to find the number of crimes reported in a city per day. 

 

  • In call centers, discrete random variables, such as the number of calls per hour, help manage staffing and performance.
Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in Discrete Random Variables

In probability, understanding the discrete random variable is important, but students often make errors. Here are a few common mistakes and ways to avoid them.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Confusing discrete and continuous random variables

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students frequently mistake discrete variables for continuous ones.

 

For example, thinking the number of customers (a discrete variable) is continuous. To avoid confusion, verify if the variables are countable. If it can be countable (like 1, 2, 3, …), it is discrete; if it is uncountable, then it is not discrete. 

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Not defining the random variable

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

While solving discrete random variables problems, students cannot define the random variable x. It leads to errors while solving issues. So for a random variable, the value should be well-defined.

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Confusing the sample space with the random variable

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students sometimes confuse the sample space with the random variable, as they are not the same. To avoid this confusion, understand that a random variable maps outcomes in the sample space to real numbers.

 

For example, in a coin toss, the sample space is {H, T}, and the random variable in the case of a coin toss can be X(H) = 1, and X(T) = 0

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Not verifying whether the sum of the probabilities is 1

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students list all the probabilities of each value of a random variable, but the sum is not equal to 1, which means the distribution is wrong. So, always remember that the sum of all the probabilities of every random variable should be 1.

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Errors while calculating the mean

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

When finding the mean of a discrete random variable, students make errors such as forgetting a value, using the wrong probability, or forgetting to multiply the value by the probability. So always remember that formula to find the mean, E[X] = ΣxP (X = x).

arrow-right
Max from BrightChamps Saying "Hey"

Solved Examples of Discrete Random Variables

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Let X be the number shown on a fair six-sided die. Find the probability mass function and expected value

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The probability mass function is ⅙, and the expected value is 3.5

Explanation

The probability mass function is the ratio of the number of outcomes to the total outcomes

Here, k = 1, 2, 3, 4, 5, 6

So, the sample space is {1, 2, 3, 4, 5, 6}, so |S| = 6
P(X = k) = ⅙

The expected value: E[X] = ΣxP (X = x)

Here, P(X = x) = ⅙

So, E[X] = Σx × 1/6

= (1 + 2 + 3 + 4 + 5 + 6)/6
 
= 21/6 = 3.5

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 2

Let X be the number of heads when a fair coin is tossed twice. Find the probability distribution of X.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

P(X = 0) = P(TT) = 1/4

P(X = 1) = P(HT or TH) = 1/2

P(X = 2) = P(HH) = 1/4

Explanation

The sample space = {TT, TH, HT, HH}

Assuming the number of heads observed as X, then

Outcomes TT → X = 0

Outcomes TH, HT → X = 1

Outcomes HH → X = 2

 

The probability = favorable outcomes/total outcomes. 

P(X = 0) = 1/4

P(X = 1) = 2/4 = 1/2

P(X = 2) = 1/4

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 3

Let X be the outcome when a fair die is rolled. Find the expected value.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The expected value is 3.5

Explanation

Here, P(X = k) = ⅙

k = 1, 2, 3, 4, 5, 6

The expected value is calculated using: E[X] = ΣxP (X = x) = (1 - p)x - 1 p

= 1/6 (1 + 2 + 3 + 4 + 5 + 6)

= 1/6 × 21

= 3.5

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 4

A machine fails with probability p = 0.4 per test. Let X be the number of tests until the first failure. Find P(X = 3)

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

Here, P(X = 3) = 0.144

Explanation

The formula to find the probability mass function of a geometric random variable is:

P(X = k) = (1 - p)k - 1 × p

= (1 - 0.4)2 × 0.4 

= (0.6)2 × 0.4

= 0.36 × 0.4 

= 0.144

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 5

A bookstore receives an average of 2 online orders per hour. Let X ∼ Poisson be the number of orders in an hour. Find P(X = 3)

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

Here, P(X = 3) = 0.18

Explanation

The probability mass function of a Poisson random variable, 
P(X = x) = λxe/x!

Here, λ = 2 and x = 3

P(X = x) = (2)3e-2/3!

= 8 × e-2/3 × 2 × 1, where e = 2.718

= (8 × 2.718-2) / 6

= 1.0824 / 6 

= 0.18

Max from BrightChamps Praising Clear Math Explanations
Ray Thinking Deeply About Math Problems

FAQs on Discrete Random Variable

1.What is a discrete random variable?

Math FAQ Answers Dropdown Arrow

2.How to calculate the expected value of a discrete random variable?

Math FAQ Answers Dropdown Arrow

3.Can a discrete random variable have infinite possible values?

Math FAQ Answers Dropdown Arrow

4.Can the value of a discrete random variable be a decimal?

Math FAQ Answers Dropdown Arrow

5.What are the applications of discrete random variables?

Math FAQ Answers Dropdown Arrow

6.How does learning Algebra help students in India make better decisions in daily life?

Math FAQ Answers Dropdown Arrow

7.How can cultural or local activities in India support learning Algebra topics such as Discrete Random Variable ?

Math FAQ Answers Dropdown Arrow

8.How do technology and digital tools in India support learning Algebra and Discrete Random Variable ?

Math FAQ Answers Dropdown Arrow

9.Does learning Algebra support future career opportunities for students in India?

Math FAQ Answers Dropdown Arrow
Math Teacher Background Image
Math Teacher Image

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Max, the Girl Character from BrightChamps

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
Dubai - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom