Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Overview
This chapter studies random variables and their distributions — finding probabilities, mean, variance, and using the binomial distribution.
1. Random Variables and PMF/PDF
Discrete: probability mass function P(x), with ΣP(x) = 1
Continuous: probability density function f(x), with ∫f(x)dx = 1
2. Mean and Variance
Mean μ = E(X) = Σ x P(x) (continuous: ∫ x f(x) dx)
E(X²) = Σ x² P(x)
Variance σ² = E(X²) − [E(X)]²
3. Binomial Distribution
P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ , q = 1 − p
Mean = np , Variance = npq , SD = √(npq)
Worked Examples
Example 1 — Validate a PMF and find mean
X: P(0)=0.2, P(1)=0.5, P(2)=0.3. Find E(X).
ΣP = 0.2+0.5+0.3 = 1 ✓
E(X) = 0(0.2) + 1(0.5) + 2(0.3) = 0.5 + 0.6 = 1.1
Example 2 — Variance
For the same X, find the variance.
E(X²) = 0(0.2) + 1(0.5) + 4(0.3) = 0.5 + 1.2 = 1.7
Var = 1.7 − (1.1)² = 1.7 − 1.21 = 0.49
Example 3 — Binomial probability
A fair coin is tossed 5 times. Find P(exactly 3 heads).
n=5, p=1/2, r=3
P = ⁵C₃ (1/2)³(1/2)² = 10 × 1/32 = 10/32 = 5/16
Example 4 — Binomial mean and variance
For n = 20, p = 0.1, find mean and variance.
Mean = np = 20 × 0.1 = 2
Variance = npq = 20 × 0.1 × 0.9 = 1.8
Common Mistakes
- Not checking ΣP(x) = 1 before computing.
- Using variance = E(X²) without subtracting [E(X)]².
- Mixing up p and q in the binomial formula.
- Forgetting the ⁿCᵣ coefficient.
Exam Tips & Revision Checklist
- Variance = E(X²) − μ² is faster than the deviation formula.
- Binomial: mean np, variance npq — always q < 1.
- Practised: PMF validation, mean, variance, binomial probability and parameters.
Frequently Asked Questions
A random variable assigns a real number to each outcome of a random experiment. It is discrete (countable values) or continuous (interval of values).