Chapter 8 — Random Variables and Probability Distributions
Key Concepts Quick Recap
| Distribution | Parameters | Mean | Variance | When to Use |
|---|---|---|---|---|
| Binomial B(n, p) | n trials, p success prob | np | npq | Fixed trials, two outcomes |
| Poisson P(λ) | λ = mean events | λ | λ | Rare events, large n, small p |
| Normal N(μ, σ²) | μ mean, σ² variance | μ | σ² | Continuous data, large samples |
I. Choose the Best Answer (MCQs)
1. A random variable X follows Binomial distribution with n = 5, p = 0.3. The mean of X is: (a) 0.15 (b) 1.5 (c) 3 (d) 5
2. In a Poisson distribution with parameter λ = 3, the mean and variance are both: (a) 3 and 6 (b) 3 and 3 (c) 6 and 3 (d) 9 and 3
3. The normal distribution is symmetric about: (a) Zero (b) Standard deviation (c) Mean (μ) (d) Median
4. For a standard normal distribution Z ~ N(0,1), the standard normal variate is: (a) Z = (X − σ)/μ (b) Z = (X − μ)/σ (c) Z = μ/σ (d) Z = X × σ
5. The area under the normal curve between μ−σ and μ+σ is approximately: (a) 50% (b) 68.27% (c) 95.45% (d) 99.73%
6. Which distribution is used for modelling the number of defects per unit in quality control? (a) Binomial (b) Normal (c) Poisson (d) Uniform
7. If X ~ B(n=10, p=0.5), the probability of getting exactly 5 successes, P(X=5), uses: (a) P(X=5) = λ⁵e⁻λ/5! (b) P(X=5) = C(10,5)(0.5)⁵(0.5)⁵ (c) P(X=5) = 1/10 (d) Area under normal curve
II. Short Answer Questions
Exam Checklist
- Binomial: n fixed, 2 outcomes, constant p, independent; Mean=np, Var=npq.
- Poisson: λ = np, mean = variance = λ; for rare events.
- Normal: bell-shaped, symmetric; μ±1σ=68.27%, μ±2σ=95.45%, μ±3σ=99.73%.
- Z = (X−μ)/σ — standardise to use Z-table.
- Poisson approximates Binomial when n large, p small, λ = np moderate.