Probability and Statistics · Discrete distributions

distribution	counts	parameters	mean
Bernoulli	one yes/no trial	p	p
Binomial	successes in n trials	n, p	np
Geometric	trials until the first success	p	1/p
Poisson	events in a fixed interval	λ	λ

Poisson is also the limit of Binomial(n, p) when n is large and p is small, with λ = np.

In scipy.stats, every distribution shares one interface: pmf (or pdf), cdf, sf (= 1 − cdf), mean, var, rvs (random samples).

from scipy import stats

b = stats.binom(n=10, p=0.3)
print("P(exactly 3 of 10) =", round(b.pmf(3), 4), "| P(at most 3) =", round(b.cdf(3), 4), "| mean", b.mean())

g = stats.geom(p=1 / 6)
print("P(first six on roll 4) =", round(g.pmf(4), 4), "| mean rolls", g.mean())

pois = stats.poisson(mu=4)          # e.g. 4 emails per hour
print("P(0 emails) =", round(pois.pmf(0), 4), "| P(more than 6) =", round(pois.sf(6), 4))

big = stats.binom(n=1000, p=0.004)  # Poisson approximation, lambda = 4
print("binomial vs poisson at k=2:", round(big.pmf(2), 5), round(pois.pmf(2), 5))
print("samples:", b.rvs(size=8, random_state=0))

Output:

P(exactly 3 of 10) = 0.2668 | P(at most 3) = 0.6496 | mean 3.0
P(first six on roll 4) = 0.0965 | mean rolls 6.0
P(0 emails) = 0.0183 | P(more than 6) = 0.1107
binomial vs poisson at k=2: 0.14638 0.14653
samples: [3 4 3 3 3 3 3 5]