Probability Distributions
Probability Distributions covers random variables and the three standard models: binomial, Poisson and normal. The exam tests probability calculations, mean and variance, and choosing the correct distribution from the wording of a problem.
Core rules
- A discrete random variable has a probability mass function with probabilities summing to 1; a continuous random variable uses a density function with total area 1.
- Binomial: fixed n independent trials, constant p, two outcomes. P(X=r) = nCr pʳ qⁿ⁻ʳ, mean = np, variance = npq.
- Poisson: counts of rare events in a fixed interval. P(X=r) = e^{−m}mʳ/r!, mean = variance = m.
- Normal: symmetric bell curve with mean μ and standard deviation σ. Convert to z = (x−μ)/σ and use standard normal tables; P(Z<0) = 0.5.
- Binomial tends to Poisson when n is large and p is small, with m = np.
Worked example
If X ~ B(5, 0.2), find P(X=2). P(X=2) = 5C2 (0.2)²(0.8)³. 5C2 = 10; (0.2)² = 0.04; (0.8)³ = 0.512. P(X=2) = 10 × 0.04 × 0.512 = 0.2048. Mean = np = 1 and variance = npq = 5(0.2)(0.8) = 0.8.
Traps
Using nPr instead of nCr; mixing up mean np with variance npq; using the Poisson formula when trials are not independent or p is not small; and forgetting that a normal probability is an area, so P(X=x) = 0 for a continuous variable.