←Back to the bookChapter 37 · Discrete Distributions

Chapter 37 · Probability Distributions

Discrete Distributions 🔔

A handful of named distributions describe most counting problems you will ever meet. This notebook builds the binomial, geometric, Poisson, and hypergeometric distributions, checks each mean and variance by simulation, watches the binomial become Poisson, and ends with the Bernoulli likelihood that powers logistic regression.

Statistics, Data Science and AI: A Visual Handbook · John Fisher · 2026

⚙️ Setup

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
rng = np.random.default_rng(37)
plt.rcParams.update({"figure.dpi":110,"font.size":11,"axes.spines.top":False,"axes.spines.right":False})
TEAL="#0d9488"; PINK="#db2777"; AMBER="#d97706"
print("ready")

ready

DEMO 1 · BERNOULLI & BINOMIAL

Counting successes in n trials

A Bernoulli trial is a single yes/no event with success probability p. A binomial counts the successes in n independent Bernoulli trials. Its mean is np and its variance is np(1-p).

n, p = 10, 0.3
k = np.arange(0, n+1)
pmf = stats.binom.pmf(k, n, p)
print(f"mean np        = {n*p:.3f}")
print(f"variance np(1-p) = {n*p*(1-p):.3f}")

sim = rng.binomial(n, p, size=200_000)
print(f"simulated mean = {sim.mean():.3f},  simulated var = {sim.var():.3f}")

mean np        = 3.000
variance np(1-p) = 2.100
simulated mean = 2.998,  simulated var = 2.096

fig,ax=plt.subplots(figsize=(7,3.2))
ax.bar(k, pmf, color=TEAL, edgecolor="white")
ax.axvline(n*p, color=PINK, ls="--", lw=2, label=f"mean = np = {n*p:.1f}")
ax.set_xticks(k); ax.set_xlabel("number of successes k"); ax.set_ylabel("P(X = k)")
ax.set_title(f"Binomial(n={n}, p={p}) PMF"); ax.legend()
plt.tight_layout(); plt.show()

The binomial PMF is C(n, k) p^k (1-p)^(n-k): the number of ways to place k successes, times the probability of each arrangement. With n=10, p=0.3 the distribution centers on np = 3, and the simulated mean and variance match the formulas.

DEMO 2 · GEOMETRIC

How long until the first success?

The geometric distribution counts the number of trials up to and including the first success. Its PMF is (1-p)^(k-1) p, and its mean is 1/p, the average wait.

p = 0.2
k = np.arange(1, 26)
pmf = stats.geom.pmf(k, p)
print(f"mean 1/p = {1/p:.1f} trials to the first success")
sim = rng.geometric(p, size=200_000)
print(f"simulated mean = {sim.mean():.3f}")

mean 1/p = 5.0 trials to the first success
simulated mean = 5.012

fig,ax=plt.subplots(figsize=(7,3))
ax.bar(k, pmf, color=TEAL, edgecolor="white")
ax.axvline(1/p, color=PINK, ls="--", lw=2, label=f"mean = 1/p = {1/p:.0f}")
ax.set_xlabel("trial of first success k"); ax.set_ylabel("P(X = k)")
ax.set_title(f"Geometric(p={p}) PMF: a long right tail"); ax.legend()
plt.tight_layout(); plt.show()

The geometric distribution is "memoryless" and skewed right: with p = 0.2 the single most likely outcome is success on the first try, yet the long tail pulls the average wait up to 1/p = 5 trials. Waiting for a rare event takes longer, on average, than intuition suggests.

DEMO 3 · POISSON

Counts of rare events, and the binomial limit

The Poisson distribution counts events in a fixed interval when they happen at a steady average rate lambda. Remarkably, mean = variance = lambda, and it is the limit of a binomial with large n and small p.

lam = 3.0
k = np.arange(0, 12)
pois = stats.poisson.pmf(k, lam)
print(f"Poisson: mean = variance = lambda = {lam}")

# binomial -> Poisson: large n, small p, with n*p = lambda
n, p = 1000, lam/1000
binom = stats.binom.pmf(k, n, p)
print(f"max |binom(n=1000,p=0.003) - Poisson(3)| = {np.abs(binom-pois).max():.5f}")

Poisson: mean = variance = lambda = 3.0
max |binom(n=1000,p=0.003) - Poisson(3)| = 0.00034

fig,ax=plt.subplots(figsize=(7,3.2))
ax.bar(k-0.18, pois, width=0.36, color=TEAL, label="Poisson(3)")
ax.bar(k+0.18, binom, width=0.36, color=AMBER, label="Binomial(1000, 0.003)")
ax.set_xticks(k); ax.set_xlabel("number of events k"); ax.set_ylabel("P(X = k)")
ax.set_title("A binomial with large n and small np becomes Poisson"); ax.legend()
plt.tight_layout(); plt.show()

The two distributions are nearly identical: when successes are rare but trials are many, the binomial collapses onto the Poisson with lambda = np. This is why the Poisson models call arrivals, typos per page, or website hits per minute, all "many chances, each unlikely".

DEMO 4 · HYPERGEOMETRIC

Sampling without replacement

When you draw without replacement, trials are no longer independent and the binomial no longer applies. The hypergeometric distribution handles it: draw N items from a population of M with K 'successes'.

M, K, N = 50, 5, 10        # 50 items, 5 defective, draw 10 without replacement
k = np.arange(0, 6)
hyper = stats.hypergeom.pmf(k, M, K, N)
binom = stats.binom.pmf(k, N, K/M)    # the with-replacement approximation
print("defects   hypergeometric   binomial(p=0.1)")
for ki,h,b in zip(k,hyper,binom):
    print(f"   {ki}        {h:.4f}          {b:.4f}")
print(f"\nP(exactly 1 defective), no replacement = {hyper[1]:.4f}")

defects   hypergeometric   binomial(p=0.1)
   0        0.3106          0.3487
   1        0.4313          0.3874
   2        0.2098          0.1937
   3        0.0442          0.0574
   4        0.0040          0.0112
   5        0.0001          0.0015

P(exactly 1 defective), no replacement = 0.4313

The hypergeometric (without replacement) and binomial (with replacement) agree roughly but not exactly: removing items changes the odds for the next draw. When the population is large relative to the sample, the difference shrinks and the simpler binomial is a fine approximation.

DEMO 5 · THE BERNOULLI LIKELIHOOD

How a classifier learns p

Logistic regression and every probabilistic classifier rest on the Bernoulli distribution. Given data, the value of p that maximizes the Bernoulli likelihood is exactly the observed success rate.

true_p = 0.7
data = rng.binomial(1, true_p, size=200)      # 200 Bernoulli trials
k = data.sum()
print(f"observed: {k} successes in 200 -> sample proportion = {k/200:.3f}")

# log-likelihood of p given the data, over a grid
grid = np.linspace(0.01, 0.99, 400)
loglik = k*np.log(grid) + (200-k)*np.log(1-grid)
p_hat = grid[np.argmax(loglik)]
print(f"maximum-likelihood estimate p_hat = {p_hat:.3f}  (matches the proportion)")

observed: 128 successes in 200 -> sample proportion = 0.640
maximum-likelihood estimate p_hat = 0.639  (matches the proportion)

fig,ax=plt.subplots(figsize=(7,3))
ax.plot(grid, loglik, color=TEAL, lw=2)
ax.axvline(p_hat, color=PINK, ls="--", lw=2, label=f"p_hat = {p_hat:.2f}")
ax.set_xlabel("p"); ax.set_ylabel("log-likelihood")
ax.set_title("The Bernoulli log-likelihood peaks at the sample proportion"); ax.legend()
plt.tight_layout(); plt.show()

Maximizing the Bernoulli likelihood is the same as minimizing cross-entropy loss, the negative log-likelihood from Chapter 35. A logistic-regression classifier does exactly this: it tunes its parameters so the predicted Bernoulli probability best matches the observed labels. The humble coin-flip distribution is the statistical heart of binary classification.

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✅ What you built

You constructed the four workhorse discrete distributions, confirmed their means and variances by simulation, watched a binomial become Poisson, compared drawing with and without replacement, and maximized a Bernoulli likelihood the way a classifier does. Each "named" distribution is just a reusable recipe for a common kind of count.

Statistics, Data Science and AI: A Visual Handbook · © 2026 John Fisher

←Back to the bookChapter 37 · Discrete Distributions