←Back to the bookChapter 42 · Joint, Marginal & Conditional Densities

Chapter 42 · Mathematical Statistics

Joint, Marginal & Conditional Densities 🎓

Two variables at once. This notebook builds a joint distribution, collapses it to marginals, conditions to get one variable given another, tests independence, and explores the bivariate normal whose conditional mean is a straight line, the seed of regression.

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

⚙️ Setup

import numpy as np
import matplotlib.pyplot as plt
rng = np.random.default_rng(42)
plt.rcParams.update({"figure.dpi":110,"font.size":11,"axes.spines.top":False,"axes.spines.right":False})
VIOLET="#7c3aed"; PINK="#db2777"; TEAL="#0d9488"
print("ready")

ready

DEMO 1 · THE JOINT DISTRIBUTION

Two variables, one table

A joint distribution gives the probability of each PAIR of outcomes. Here is a joint PMF over weather (rain/clear) and commute (late/on-time) as a 2x2 table of probabilities that sum to 1.

# rows = weather, cols = commute
joint = np.array([[0.20, 0.10],    # rain:  [late, on-time]
                  [0.15, 0.55]])   # clear: [late, on-time]
print("joint P(weather, commute):")
print(joint)
print(f"sums to 1? {joint.sum():.2f}")
print(f"P(rain AND late) = {joint[0,0]:.2f}")

joint P(weather, commute):
[[0.2  0.1 ]
 [0.15 0.55]]
sums to 1? 1.00
P(rain AND late) = 0.20

fig,ax=plt.subplots(figsize=(4.6,3.6))
im=ax.imshow(joint, cmap="Purples", vmin=0, vmax=0.6)
for i in range(2):
    for j in range(2):
        ax.text(j,i,f"{joint[i,j]:.2f}",ha="center",va="center",fontweight="bold",color="#1a2138")
ax.set_xticks([0,1],["late","on-time"]); ax.set_yticks([0,1],["rain","clear"])
ax.set_title("Joint PMF P(weather, commute)"); plt.colorbar(im,fraction=0.046); plt.tight_layout(); plt.show()

The joint distribution captures everything about how the two variables behave together. Each cell is the probability of one specific combination, and the four cells sum to 1. From this single table we can recover every other quantity in this chapter.

DEMO 2 · MARGINAL DISTRIBUTIONS

Collapsing to one variable

A marginal distribution describes one variable on its own, obtained by SUMMING the joint over the other (for densities, integrating it out).

p_weather = joint.sum(axis=1)   # sum over commute
p_commute = joint.sum(axis=0)   # sum over weather
print(f"marginal P(weather): rain={p_weather[0]:.2f}, clear={p_weather[1]:.2f}")
print(f"marginal P(commute): late={p_commute[0]:.2f}, on-time={p_commute[1]:.2f}")

marginal P(weather): rain=0.30, clear=0.70
marginal P(commute): late=0.35, on-time=0.65

Summing each row gives the marginal for weather (30% rain, 70% clear); summing each column gives the marginal for commute (35% late, 65% on-time). "Marginal" is literally the total you would write in the margin of the table, the variable on its own, ignoring the other.

DEMO 3 · CONDITIONAL & INDEPENDENCE

One variable given another

The conditional distribution f(y|x) = f(x,y) / f_X(x) re-normalizes one row (or column) to sum to 1. If conditioning changes nothing, the variables are independent.

# P(commute | rain): divide the rain row by P(rain)
p_commute_given_rain = joint[0] / p_weather[0]
print(f"P(commute | rain):  late={p_commute_given_rain[0]:.3f}, on-time={p_commute_given_rain[1]:.3f}")
print(f"compare marginal P(commute): late={p_commute[0]:.3f}  -> differs, so DEPENDENT")

# independence check: is joint == outer product of marginals?
indep = np.outer(p_weather, p_commute)
print(f"\nwould-be-independent joint:\n{indep.round(3)}")
print(f"matches actual joint? {np.allclose(joint, indep)}")

P(commute | rain):  late=0.667, on-time=0.333
compare marginal P(commute): late=0.350  -> differs, so DEPENDENT

would-be-independent joint:
[[0.105 0.195]
 [0.245 0.455]]
matches actual joint? False

Given rain, the chance of being late jumps from the unconditional 35% to 67%. Because conditioning changed the distribution, weather and commute are dependent. The formal test: they are independent only if the joint equals the product of the marginals, and here it does not.

DEMO 4 · THE BIVARIATE NORMAL

Correlated continuous variables

The bivariate normal is the joint distribution of two correlated Gaussians. Its shape is set by a covariance matrix; the correlation rho tilts the cloud.

mean = [0, 0]
rho = 0.75
cov = [[1, rho],[rho, 1]]
xy = rng.multivariate_normal(mean, cov, size=5000)
print(f"target correlation rho = {rho}")
print(f"sample covariance matrix:\n{np.cov(xy.T).round(3)}")
print(f"sample correlation = {np.corrcoef(xy.T)[0,1]:.3f}")

target correlation rho = 0.75
sample covariance matrix:
[[0.999 0.726]
 [0.726 0.976]]
sample correlation = 0.735

fig,ax=plt.subplots(figsize=(4.6,4.6))
ax.scatter(xy[:,0], xy[:,1], s=5, alpha=0.25, color=VIOLET)
xs=np.linspace(-3,3,50); ax.plot(xs, rho*xs, color=PINK, lw=2.5, label="E[Y|X=x] = rho*x")
ax.set_aspect("equal"); ax.set_xlabel("X"); ax.set_ylabel("Y")
ax.set_title(f"Bivariate normal (rho={rho})"); ax.legend()
plt.tight_layout(); plt.show()

The cloud tilts because X and Y are correlated. The pink line is the conditional mean E[Y | X = x], and for the bivariate normal it is exactly a straight line through the data, rho*x here. That straight conditional mean is the theoretical seed of linear regression (Part XII).

DEMO 5 · JOINT vs CONDITIONAL IN ML

Generative and discriminative models

The joint-vs-conditional split is one of the deepest divides in machine learning. A generative model learns the joint P(x, y); a discriminative model learns only the conditional P(y | x).

# from the joint we can get BOTH a discriminative conditional and a generative joint
print("Discriminative view  P(commute | weather):")
for i,w in enumerate(["rain","clear"]):
    cond = joint[i]/joint[i].sum()
    print(f"   given {w:5s}: late={cond[0]:.2f}, on-time={cond[1]:.2f}")
print("\nGenerative view  P(weather, commute): the full joint table above.")
print("Discriminative (logistic reg, LLMs) models P(y|x); generative (naive Bayes, VAEs) models the joint.")

Discriminative view  P(commute | weather):
   given rain : late=0.67, on-time=0.33
   given clear: late=0.21, on-time=0.79

Generative view  P(weather, commute): the full joint table above.
Discriminative (logistic reg, LLMs) models P(y|x); generative (naive Bayes, VAEs) models the joint.

A spam classifier and a language model are discriminative: they estimate P(label | features) or P(next token | context) directly. A generative model (naive Bayes, a VAE, a diffusion model) learns the full joint distribution and can sample new data from it. Both are views of the same joint distribution you built in Demo 1.

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✅ What you built

You built a joint distribution, recovered its marginals, conditioned to get one variable given another, tested independence, explored the bivariate normal and its linear conditional mean, and saw the joint-vs-conditional split that separates generative from discriminative models. The joint distribution is the master object; everything else is derived from it.

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

←Back to the bookChapter 42 · Joint, Marginal & Conditional Densities