Lois jointes, marginales, conditionnelles

import numpy as np
from math import erf, sqrt

rng = np.random.default_rng(0)
n = 100_000
clouds = {
    "gaussian": rng.standard_normal((n, 2)),
    "uniform square": rng.uniform(-2, 2, (n, 2)),
}
disk = rng.uniform(-2, 2, (2 * n, 2))
clouds["uniform disk"] = disk[(disk ** 2).sum(axis=1) < 4][:n]   # keep points inside radius 2

bands = [(0, 0.5), (0.5, 1), (1, 1.5), (1.5, 2)]
row = lambda shares: "".join(f"{100 * s:9.1f} %" for s in shares)
print(f"{'band of x':15s}" + "".join(f"{str([a, b]):>11s}" for a, b in bands))
for name, pts in clouds.items():
    x = pts[:, 0]                  # marginal of X: forget y, count points band by band
    print(f"{name:15s}" + row([np.mean((x >= a) & (x < b)) for a, b in bands]))

Phi = lambda t: 0.5 * (1 + erf(t / sqrt(2)))
print(f"{'gaussian, exact':15s}" + row([Phi(b) - Phi(a) for a, b in bands]))

band of x         [0, 0.5]   [0.5, 1]   [1, 1.5]   [1.5, 2]
gaussian            19.0 %     15.1 %      9.1 %      4.5 %
uniform square      12.3 %     12.5 %     12.5 %     12.6 %
uniform disk        15.6 %     14.8 %     12.3 %      7.2 %
gaussian, exact     19.1 %     15.0 %      9.2 %      4.4 %

import numpy as np

rng = np.random.default_rng(0)
n = 2_000_000
x = rng.standard_normal(n)
y = 0.8 * x + 0.6 * rng.standard_normal(n)       # the cloud of the vignette

band = np.abs(x - 1.5) < 0.02                     # a thin slice around X = 1.5
print(f"points in the slice: {band.sum()}")
print(f"slice : mean {y[band].mean():.3f} (theory 1.2), sd {y[band].std():.3f} (theory 0.6)")
print(f"all y : mean {y.mean():.3f} (theory 0),   sd {y.std():.3f} (theory 1)")

# discrete: first die X and maximum M; condition on X = 2 by keeping that column only
d1, d2 = rng.integers(1, 7, n), rng.integers(1, 7, n)
col = np.maximum(d1, d2)[d1 == 2]
print("P(M = m | X = 2), m = 1..6:", [round(float(np.mean(col == v)), 3) for v in range(1, 7)])

points in the slice: 10304
slice : mean 1.196 (theory 1.2), sd 0.604 (theory 0.6)
all y : mean -0.000 (theory 0),   sd 1.000 (theory 1)
P(M = m | X = 2), m = 1..6: [0.0, 0.333, 0.167, 0.167, 0.167, 0.166]

import numpy as np
from math import pi

def joint(x, y, rho):                   # standard bivariate normal with correlation rho
    q = (x**2 - 2 * rho * x * y + y**2) / (1 - rho**2)
    return np.exp(-q / 2) / (2 * pi * np.sqrt(1 - rho**2))
phi = lambda t: np.exp(-t**2 / 2) / np.sqrt(2 * pi)   # both marginals are N(0, 1), for any rho

g = np.linspace(-3, 3, 301)
X, Y = np.meshgrid(g, g)
for rho in (0.0, 0.8):
    gap = np.abs(joint(X, Y, rho) - phi(X) * phi(Y)).max()
    print(f"rho = {rho}: max |f(x, y) - fX(x) fY(y)| on the grid = {gap:.4f}")

rng = np.random.default_rng(0)
n = 1_000_000
r, t = 2 + 0.2 * rng.standard_normal(n), rng.uniform(0, 2 * pi, n)
clouds = {"centre": rng.standard_normal((2, n)), "ring": (r * np.cos(t), r * np.sin(t))}
for name, (x, y) in clouds.items():
    ys = rng.permutation(y)             # shuffle the y's: builds the product of the marginals
    before, after = np.mean(x**2 + y**2 < 1), np.mean(x**2 + ys**2 < 1)
    print(f"{name:6s}: share with r < 1 before {before:.3f}, after shuffling {after:.3f}")

rho = 0.0: max |f(x, y) - fX(x) fY(y)| on the grid = 0.0000
rho = 0.8: max |f(x, y) - fX(x) fY(y)| on the grid = 0.1071
centre: share with r < 1 before 0.394, after shuffling 0.394
ring  : share with r < 1 before 0.000, after shuffling 0.087

import numpy as np
from math import exp, pi, sqrt

pdf = lambda y, mu: exp(-(y - mu) ** 2 / 2) / sqrt(2 * pi)   # N(mu, 1) density
prev = 0.01
for y in (2, 3):
    num = pdf(y, 2) * prev
    den = num + pdf(y, 0) * (1 - prev)          # f_Y(y), the marginal of the score
    ratio = pdf(y, 2) / pdf(y, 0)               # likelihood ratio, sick vs healthy
    print(f"score {y}: likelihood ratio {ratio:.2f}, P(sick | score) = {num / den:.4f}")

rng = np.random.default_rng(0)
n = 2_000_000
sick = rng.random(n) < prev
score = rng.standard_normal(n) + 2 * sick       # N(0, 1) if healthy, N(2, 1) if sick
band = np.abs(score - 2) < 0.05                 # condition on Y ≈ 2 with a thin band
print(f"people with a score near 2: {band.sum()}")
print(f"share of sick among them:   {sick[band].mean():.4f}")

score 2: likelihood ratio 7.39, P(sick | score) = 0.0695
score 3: likelihood ratio 54.60, P(sick | score) = 0.3555
people with a score near 2: 11647
share of sick among them:   0.0705