Module 3 · Distributions · Topic 9
When to Use Which Distribution. Binomial: fixed n independent trials with constant p. Poisson: count of events in a fixed interval with rate λ. Normal: continuous data that is symmetric and bell-shaped.
Normal Approximation to Binomial. When np ≥ 10 and n(1 − p) ≥ 10, the binomial is well-approximated by a normal distribution with μ = np and σ = √(np(1 − p)). Apply a continuity correction of ±0.5 for better accuracy.
P(X = k) = C(n,k)p^k(1-p)^(n-k) — Exact probability of k successes in n independent Bernoulli trials.z = (x - mu) / sigma — Standardizes a value by measuring standard deviations from the mean.n = 200, p = 0.45. μ = 90, σ = √(200 × 0.45 × 0.55) = √49.5 ≈ 7.04. P(X ≥ 100) ≈ P(Z ≥ (99.5 − 90)/7.04) = P(Z ≥ 1.35) ≈ 0.0885.
Discrete + fixed trials → Binomial. Discrete + rare events/rate → Poisson. Continuous + bell-shaped → Normal. Always check conditions before applying a distribution.
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