Bernoulli→Binomial→Poisson→Normal

0.20
10
λ = np = 2.00

Bernoulli(p)one trial

P(X = 1) = p,   P(X = 0) = 1 − p
this trial is trial 1 of the binomial strip
→
add up
n trials

Binomial(n, p)X = number of successes

P(X = k) = C(n, k) pk (1 − p)n−k,   k = 0, …, n
a successthis run's X
→
n → ∞
p → 0
np = λ fixed

Poisson(λ)X = events in the interval

P(X = k) = e−λ λk / k!,   k = 0, 1, 2, …
Poisson(λ)Binomial(n, p), same mean

Why: add up small effects

Each effect is a
Add up 1
One effect on its own can look like anything. Add up enough independent effects and the total follows the violet normal curve: the Central Limit Theorem. That is why the quantities on the left are close to normal: each is the sum of many small, independent nudges.
Standard normal table: area to the LEFT of z
or click any cell
X ~ N(μ, σ²) with
Try: