Distribution formulas
Binomial
μ = np, σ² = np(1−p). Use when counting successes in n independent yes/no trials with constant success probability p.
Poisson
μ = λ, σ² = λ. Use for rare events occurring at average rate λ over a fixed interval (calls per hour, defects per page).
Normal
For continuous data, P(X = exact value) = 0; we report the PDF and the cumulative P(X ≤ x).
FAQ
When can I approximate binomial by normal?
When np ≥ 10 and n(1−p) ≥ 10 (the so-called success-failure condition). At that scale, the binomial is well-approximated by Normal(np, np(1−p)), which speeds up CDF computation.
When can I approximate binomial by Poisson?
When n is large and p is small with np = λ moderate (typically n ≥ 20 and p ≤ 0.05). The binomial PMF then converges to the Poisson PMF.
How accurate is the normal CDF here?
The implementation uses the Abramowitz-Stegun approximation accurate to ~7 decimal places, which is well beyond what's needed for any practical calculation.