Random Variables: Discrete and Continuous
So far, we have talked about probability in terms of events. To move further in statistics, we need a way to represent outcomes using numbers.
This is where random variables come into play. They form the bridge between probability and numerical analysis.
What Is a Random Variable?
A random variable is a variable whose value depends on the outcome of a random experiment.
In simple words, it assigns a numerical value to each possible outcome.
Random variables allow us to apply mathematical tools to probability problems.
Simple Example
Consider tossing a coin:
- If Heads occurs → assign value 1
- If Tails occurs → assign value 0
The value obtained (0 or 1) is a random variable because it depends on chance.
Types of Random Variables
Random variables are mainly classified into two types:
- Discrete Random Variables
- Continuous Random Variables
Discrete Random Variables
A discrete random variable takes only specific, countable values. These values are usually whole numbers.
There are gaps between possible values, meaning not every value within a range is possible.
Numerical Example (Discrete)
Let X represent the number of heads obtained when tossing two coins.
Possible values of X:
- 0 (no heads)
- 1 (one head)
- 2 (two heads)
Here, X can take only these three values, so it is a discrete random variable.
Real-World Examples (Discrete)
- Number of students in a class
- Number of defective products
- Number of customers visiting a store
Continuous Random Variables
A continuous random variable can take any value within a given range. These values are measurable and can include decimals.
There are no gaps between possible values.
Numerical Example (Continuous)
Let X represent the time (in minutes) taken to complete a task.
X could be:
- 5 minutes
- 5.2 minutes
- 5.25 minutes
Because X can take infinitely many values within a range, it is a continuous random variable.
Real-World Examples (Continuous)
- Height of a person
- Weight of a package
- Temperature of a room
- Time taken to finish a race
Key Differences Between Discrete and Continuous
| Aspect | Discrete Random Variable | Continuous Random Variable |
|---|---|---|
| Possible values | Countable values | Infinite values within a range |
| Nature | Usually whole numbers | Measured values |
| Examples | Number of calls, number of items | Height, weight, time |
Why Random Variables Are Important
Random variables allow us to:
- Represent random outcomes numerically
- Apply probability formulas
- Build probability distributions
- Model real-world uncertainty
Almost all advanced topics in statistics and data science are built on random variables.
Quick Check
Is the number of emails received in a day a discrete or continuous random variable?
Practice Quiz
Question 1:
Which of the following is a continuous random variable?
- A. Number of students
- B. Time taken to complete a task
- C. Number of goals scored
Question 2:
Can a discrete random variable take decimal values?
Question 3:
Which type of random variable is used for measuring height?
Mini Practice
Classify the following as discrete or continuous:
- Number of cars passing a signal in an hour
- Amount of rainfall in a day
What’s Next
In the next lesson, we will study the Normal Distribution, one of the most important probability distributions in statistics.