Probability Distributions

• Random variables — discrete vs continuous — and their probability distributions.
• For discrete RV: PMF p(x) = P(X = x) with Σp(x) = 1.
• Expectation E(X) = Σ x·p(x) and variance Var(X) = E(X²) − [E(X)]².
• Binomial distribution B(n,p): conditions, formula C(n,r)p^r(1−p)^(n−r), mean np, variance np(1−p).
• Poisson as limit of binomial (optional JEE) and normal approximation intuition for large n.

Level 1 — Foundations

Verbal: A probability distribution lists how probability is spread over values of a random variable X.

Discrete RV: Takes countable values. PMF p(xᵢ) ≥ 0, Σ p(xᵢ) = 1.

Cumulative distribution: F(x) = P(X ≤ x) — step function for discrete case.

Expectation (mean): E(X) = Σ xᵢ p(xᵢ) — long-run average.

Variance: Var(X) = E[(X − μ)²] = E(X²) − μ². Standard deviation σ = √Var(X).

Example — fair die: X ∈ {1,…,6}, p(x) = 1/6 each; E(X) = 3.5; Var(X) = 35/12.

Level 2 — JEE / NEET depth

Binomial B(n, p): n independent Bernoulli trials, success prob p. X = number of successes.

P(X = r) = C(n,r) p^r (1−p)^(n−r), r = 0,1,…,n.

Conditions (memorise):

1. Fixed n trials.
2. Two outcomes per trial (success/failure).
3. Constant p.
4. Independent trials.

Mean & variance: E(X) = np, Var(X) = np(1−p).

Bernoulli: n = 1 special case.

Poisson (λ): P(X = k) = e^(−λ) λ^k / k! — models rare events; E(X) = Var(X) = λ.

Normal approximation: When n large, np and n(1−p) both > 5, binomial ≈ Normal(np, np(1−p)) with continuity correction for discrete → continuous.

Binomial — exactly 2 heads in 5 fair tosses

X ~ B(5, 0.5). Find P(X = 2).

Step 1 — n = 5, p = 0.5, r = 2.
Step 2 — P(X=2) = C(5,2)(0.5)²(0.5)³ = 10 × (1/32) = 10/32 = 5/16.
Step 3 — Mean check: np = 2.5 — r = 2 is near centre of distribution.

Expectation of discrete distribution

X takes values 0,1,2 with P(0)=0.2, P(1)=0.5, P(2)=0.3. Find E(X) and Var(X).

Step 1 — E(X) = 0(0.2) + 1(0.5) + 2(0.3) = 0 + 0.5 + 0.6 = 1.1.
Step 2 — E(X²) = 0²(0.2) + 1²(0.5) + 2²(0.3) = 0 + 0.5 + 1.2 = 1.7.
Step 3 — Var(X) = 1.7 − (1.1)² = 1.7 − 1.21 = 0.49.
Step 4 — σ = √0.49 = 0.7.

• Write conditions for X ~ B(n,p).
• If X ~ B(10, 0.3), find E(X) and Var(X).
• Find P(X ≥ 1) for X ~ B(3, 0.5).
• Why is E(die roll) = 3.5 not an observable outcome?
• Stretch: Show Var(X) = E(X²) − [E(X)]² from definition.

NCERT Chapter 13 link: Random variables bridge probability to statistics. Binomial distribution assumes fixed n independent trials — verify conditions before applying formula. Mean np interpretation: expected successes in n trials.

Exam connections: "At least one" problems use complement: P(X ≥ 1) = 1 − P(X = 0) = 1 − (1−p)ⁿ — faster than summing. Variance np(1−p) questions sometimes ask standard deviation — take square root at end. Board may ask to construct PMF table from word problem first.

Study strategy: For binomial, identify n (trials) and p (success per trial) before choosing r. Distinguish discrete PMF from continuous PDF — Class 12 focuses discrete binomial primarily. Verify E(X) and Var(X) formulas by expanding Σ on small n manually once.

When studying Probability Distributions within Probability, start by listing every formula and definition on one page without looking at the textbook. Compare your list to NCERT — missing items indicate gaps to fix immediately. Work through at least two NCERT Examples for this section with steps written in full; examiners award method marks even when arithmetic slips.

For board exams (CBSE), long answers benefit from a clear structure: definition → explanation → diagram or formula → example → brief conclusion. Underline key terms. For JEE Main and NEET, prioritise conceptual traps and quick calculation paths; timed mixed quizzes of 10 questions after revision simulate exam pressure.

Cross-topic link: Coordinate geometry and vectors often combine with matrices; calculus links to physics kinematics problems.

Spaced revision: Review this note at 1 day, 3 days, and 7 days after first study. Attempt the Quick check questions closed-book, then open the Practice tab for graded reinforcement. Maintain an error log — repeated mistake patterns reveal whether the issue is concept, formula recall, or careless reading.

Diagram and terminology drill: For Mathematics, redraw key figures from memory and define every labelled part in one sentence. Vocabulary precision prevents mark loss in descriptive answers — use NCERT terms exactly as printed in the textbook.

Revision tip: Link this topic to adjacent Class 12 chapters before attempting mixed practice.

Open the Practice tab for graded questions on Probability Distributions.