Random Variables & Probability Distributions

Quantifying uncertain outcomes and their distributions

What is a Random Variable?

A random variable is a function that assigns a numerical value to each outcome in a sample space. It converts qualitative outcomes into quantitative values that we can analyze mathematically. For example, in coin tosses, we assign 0 to tails and 1 to heads. Random variables are denoted by capital letters like X, Y, Z.

Random Variables

Types of Random Variables

📌 Discrete Random Variables

Definition: Takes on a countable number of distinct values
Examples: Number of heads in 5 coin tosses, number of defective items, test scores
Range: Can be finite or countably infinite

📌 Continuous Random Variables

Definition: Takes on any value in an interval or range (uncountably infinite)
Examples: Height, weight, temperature, time duration
Range: Values form a continuum
Note: Continuous RVs are not part of Class 12 curriculum but important to know

Notation

Random Variable Notation

X: Represents the random variable

X = x: The event that random variable X takes the value x

P(X = x): Probability that X equals x

E(X): Expected value (mean) of X

Var(X): Variance of X

Probability Distribution

Definition

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes for a random variable. For a discrete random variable, it lists all possible values and their corresponding probabilities.

Requirements for a Probability Distribution

P(X = xᵢ) ≥ 0 for all values of X
ΣP(X = xᵢ) = 1 (sum of all probabilities equals 1)
0 ≤ P(X = xᵢ) ≤ 1 for each value

Probability Distribution Table

A probability distribution is often presented as a table:

X	x₁	x₂	x₃	...	xₙ
P(X)	p₁	p₂	p₃	...	pₙ

Worked Examples

Example 1: Coin Toss Distribution

Problem: Two coins are tossed. Let X = number of heads. Construct the probability distribution.

Solution:

Sample Space: {HH, HT, TH, TT}

X can be 0, 1, or 2

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

P(X = 1) = P(HT, TH) = 2/4 = 1/2

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

X	0	1	2
P(X)	1/4	1/2	1/4

Example 2: Die Roll Distribution

Problem: A die is rolled. Let X = outcome. Is this a valid probability distribution?

Solution:

X ∈ {1, 2, 3, 4, 5, 6}

Each outcome: P(X = i) = 1/6

Sum: 6 × (1/6) = 1 ✓

Each probability: 0 ≤ 1/6 ≤ 1 ✓

Yes, this is a valid probability distribution (uniform distribution)

Mean and Variance of Probability Distribution

Expected Value (Mean)

Expected Value

E(X) = μ = Σ xᵢ × P(X = xᵢ)

Weighted average of all possible values

Variance

Variance (Method 1)

Var(X) = σ² = Σ(xᵢ - μ)² × P(X = xᵢ)

Variance from definition

Variance (Method 2 - Shortcut)

Var(X) = σ² = E(X²) - [E(X)]²

Easier calculation method

Standard Deviation

σ = √[Var(X)]

Square root of variance

Worked Examples

Example 1: Calculate E(X) and Var(X)

Problem: For the probability distribution:

X	0	1	2
P(X)	1/4	1/2	1/4

Solution:

E(X) = 0 × (1/4) + 1 × (1/2) + 2 × (1/4) = 0 + 1/2 + 1/2 = 1

E(X²) = 0² × (1/4) + 1² × (1/2) + 2² × (1/4) = 0 + 1/2 + 1 = 3/2

Var(X) = E(X²) - [E(X)]² = 3/2 - 1² = 3/2 - 1 = 1/2

σ = √(1/2) ≈ 0.707

Example 2: Dice Roll Mean and Variance

Problem: A fair die is rolled. Find E(X) and Var(X).

Solution:

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

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

E(X²) = 1²(1/6) + 2²(1/6) + ... + 6²(1/6) = 91/6

Var(X) = 91/6 - (3.5)² = 91/6 - 12.25 ≈ 2.917

Properties of Expectation and Variance

E(aX + b): = aE(X) + b (linearity of expectation)
Var(aX + b): = a²Var(X) (variance with constant)
E(X + Y): = E(X) + E(Y) (for any X, Y)
Var(X + Y): = Var(X) + Var(Y) + 2Cov(X,Y)
If X and Y independent: Var(X + Y) = Var(X) + Var(Y)

📋 Board Exam Tips

Q: Define random variables and probability distribution

Q: Construct probability distribution from sample space

Q: Calculate E(X) and Var(X) from distributions

Common Mistake: Forgetting to verify that all probabilities sum to 1

Helpful Tip: Use the shortcut formula Var(X) = E(X²) - [E(X)]² for easier calculation

🎯 JEE Advanced Tips

Key Insight: Understanding mean and variance of distributions is crucial for statistics

Common Pattern: Questions often ask for E(X), E(X²), Var(X) sequentially

Speed Technique: Memorize linearity properties for quick calculations

Integration: This topic connects to Chapter 14 (Mathematical Expectation) in some curricula

📐 Complete Formula Sheet

Expected Value

E(X) = Σ xᵢ × P(X = xᵢ)

Mean of distribution

Variance (Definition)

Var(X) = Σ(xᵢ - μ)² × P(X = xᵢ)

Spread from mean

Variance (Shortcut)

Var(X) = E(X²) - [E(X)]²

Easier calculation

Standard Deviation

σ = √[Var(X)]

Square root of variance

E(aX + b)

E(aX + b) = aE(X) + b

Linearity of expectation

Var(aX + b)

Var(aX + b) = a²Var(X)

Variance with constants