Class 12 Statistics Notes · GSEB

Random Variable

Random Variable — understand the meaning, types, and notation of discrete and continuous random variables. GSEB Class 12 Commerce Statistics notes with examples and key exam points.

Last updated: 22 Sep 2026

Notes

What is a Random Variable?

Random Variable

A random variable is a function that assigns a numerical value to each outcome of a random experiment. It converts qualitative outcomes (like 'head' or 'tail') into numbers we can compute with.

Think of it as a scoreboard

Just like a cricket scoreboard assigns numbers to outcomes (runs scored, wickets taken), a random variable assigns numbers to the outcomes of any random experiment. The experiment gives outcomes; the random variable gives them numerical meaning.

Discrete vs Continuous Random Variables

Random variables come in two flavours. The type determines how we compute probabilities — tables for discrete, curves for continuous.

Two Types of Random Variables
Aspect	Discrete Random Variable	Continuous Random Variable
Values	Countable (0, 1, 2, ...)	Any value in a range (150.2, 155.7, ...)
Probability	P(X = x) exists and is meaningful	P(X = x) = 0 for any exact point; use intervals
Distribution	Table or formula	Density curve (area under curve)
Example	Number of heads in 3 tosses	Height of a student
Computation	Sum of individual probabilities	Integration (area under curve)

Discrete — Bar Chart

X = heads in 3 coin tosses

Continuous — Smooth Curve

X = height of students

Real-Life Examples

Click any example to see how the random variable maps outcomes to numbers.

Discrete Examples

Continuous Examples

Notation

Discrete Notation

P(X = x) — probability that X takes the exact value x.

Example: P(X = 2) = 0.375 means there is a 37.5% chance of getting exactly 2 heads.

Continuous Notation

P(a < X < b) — probability that X falls between a and b.

Example: P(155 < X < 165) — height between 155 cm and 165 cm.

Key difference

For discrete variables, P(X = x) gives a positive number. For continuous variables, P(X = x) = 0 for any exact value — because there are infinitely many possible values, the chance of hitting one exact point is zero. Always use intervals for continuous variables.

Key Takeaways

A random variable converts outcomes of a random experiment into numerical values.
Discrete random variables take countable values (0, 1, 2, ...) — use probability tables.
Continuous random variables take any value in a range — use density curves and intervals.
For discrete: P(X = x) exists and is meaningful. For continuous: P(X = x) = 0, use P(a < X < b).
This chapter focuses on discrete random variables and their probability distributions.

Practice

A die is rolled once. Is X = "number on the die" discrete or continuous? List its possible values.
The temperature in Ahmedabad today is measured. Is this discrete or continuous? Why?
A bag contains 5 red and 3 blue balls. Two balls are drawn. Let X = number of red balls. Is X discrete or continuous? What values can X take?