Statistical Distributions Visualization

This document provides practical examples for visualizing probability distributions, including PDFs, CDFs, and comparisons across distribution families.

Setup

import matplotlib.pyplot as plt
import numpy as np
from scipy import stats

---

Normal Distribution

1. Standard Normal

x = np.linspace(-4, 4, 200)

fig, axes = plt.subplots(1, 2, figsize=(12, 4))

# PDF
axes[0].plot(x, stats.norm.pdf(x), 'b-', linewidth=2)
axes[0].fill_between(x, stats.norm.pdf(x), alpha=0.3)
axes[0].set_title('Standard Normal PDF')
axes[0].set_xlabel('x')
axes[0].set_ylabel('f(x)')
axes[0].grid(alpha=0.3)

# CDF
axes[1].plot(x, stats.norm.cdf(x), 'r-', linewidth=2)
axes[1].set_title('Standard Normal CDF')
axes[1].set_xlabel('x')
axes[1].set_ylabel('F(x)')
axes[1].grid(alpha=0.3)

plt.tight_layout()
plt.show()

2. Varying Parameters

x = np.linspace(-10, 10, 200)

fig, axes = plt.subplots(1, 2, figsize=(12, 4))

# Varying mean
for mu in [-2, 0, 2]:
    axes[0].plot(x, stats.norm(mu, 1).pdf(x), label=f'μ={mu}')
axes[0].set_title('Effect of Mean (σ=1)')
axes[0].legend()
axes[0].grid(alpha=0.3)

# Varying std
for sigma in [0.5, 1, 2]:
    axes[1].plot(x, stats.norm(0, sigma).pdf(x), label=f'σ={sigma}')
axes[1].set_title('Effect of Std Dev (μ=0)')
axes[1].legend()
axes[1].grid(alpha=0.3)

plt.tight_layout()
plt.show()

3. Normal Distribution Dashboard

mu, sigma = 2, 1.5
x = np.linspace(-4, 8, 200)
rv = stats.norm(mu, sigma)

fig, axes = plt.subplots(2, 2, figsize=(12, 10))

# PDF
axes[0, 0].plot(x, rv.pdf(x), 'b-', linewidth=2)
axes[0, 0].fill_between(x, rv.pdf(x), alpha=0.3)
axes[0, 0].axvline(mu, color='red', linestyle='--', label=f'μ={mu}')
axes[0, 0].set_title('Probability Density Function')
axes[0, 0].legend()
axes[0, 0].grid(alpha=0.3)

# CDF
axes[0, 1].plot(x, rv.cdf(x), 'g-', linewidth=2)
axes[0, 1].axhline(0.5, color='gray', linestyle=':', alpha=0.7)
axes[0, 1].axvline(mu, color='red', linestyle='--')
axes[0, 1].set_title('Cumulative Distribution Function')
axes[0, 1].grid(alpha=0.3)

# Histogram + PDF
np.random.seed(42)
samples = rv.rvs(1000)
axes[1, 0].hist(samples, bins=30, density=True, alpha=0.7, label='Samples')
axes[1, 0].plot(x, rv.pdf(x), 'r-', linewidth=2, label='PDF')
axes[1, 0].set_title('Sample Histogram vs PDF')
axes[1, 0].legend()
axes[1, 0].grid(alpha=0.3)

# Q-Q Plot
stats.probplot(samples, dist="norm", plot=axes[1, 1])
axes[1, 1].set_title('Q-Q Plot')
axes[1, 1].grid(alpha=0.3)

plt.suptitle(f'Normal Distribution (μ={mu}, σ={sigma})', fontsize=14, fontweight='bold')
plt.tight_layout()
plt.show()

---

Exponential Distribution

1. Basic Visualization

x = np.linspace(0, 8, 200)

fig, axes = plt.subplots(1, 2, figsize=(12, 4))

for lam in [0.5, 1, 2]:
    rv = stats.expon(scale=1/lam)
    axes[0].plot(x, rv.pdf(x), label=f'λ={lam}')
    axes[1].plot(x, rv.cdf(x), label=f'λ={lam}')

axes[0].set_title('Exponential PDF')
axes[0].legend()
axes[0].grid(alpha=0.3)

axes[1].set_title('Exponential CDF')
axes[1].legend()
axes[1].grid(alpha=0.3)

plt.tight_layout()
plt.show()

---

Gamma Distribution

x = np.linspace(0, 20, 200)

fig, axes = plt.subplots(1, 2, figsize=(12, 4))

# Varying shape (k)
for k in [1, 2, 5, 9]:
    rv = stats.gamma(a=k, scale=1)
    axes[0].plot(x, rv.pdf(x), label=f'k={k}, θ=1')

axes[0].set_title('Gamma PDF: Varying Shape')
axes[0].legend()
axes[0].grid(alpha=0.3)

# Varying scale (θ)
for theta in [0.5, 1, 2]:
    rv = stats.gamma(a=3, scale=theta)
    axes[1].plot(x, rv.pdf(x), label=f'k=3, θ={theta}')

axes[1].set_title('Gamma PDF: Varying Scale')
axes[1].legend()
axes[1].grid(alpha=0.3)

plt.tight_layout()
plt.show()

---

Beta Distribution

x = np.linspace(0, 1, 200)

fig, ax = plt.subplots(figsize=(10, 6))

params = [(0.5, 0.5), (2, 2), (2, 5), (5, 2), (1, 3)]
colors = plt.cm.viridis(np.linspace(0, 1, len(params)))

for (a, b), color in zip(params, colors):
    rv = stats.beta(a, b)
    ax.plot(x, rv.pdf(x), color=color, linewidth=2, label=f'α={a}, β={b}')

ax.set_title('Beta Distribution')
ax.set_xlabel('x')
ax.set_ylabel('Density')
ax.legend()
ax.grid(alpha=0.3)
plt.show()

---

Student's t-Distribution

x = np.linspace(-5, 5, 200)

fig, ax = plt.subplots(figsize=(10, 6))

# Normal for reference
ax.plot(x, stats.norm.pdf(x), 'k--', linewidth=2, label='Normal')

# t-distributions with various df
for df in [1, 2, 5, 30]:
    ax.plot(x, stats.t(df).pdf(x), linewidth=2, label=f't (df={df})')

ax.set_title("Student's t-Distribution")
ax.set_xlabel('x')
ax.set_ylabel('Density')
ax.legend()
ax.grid(alpha=0.3)
plt.show()

---

Chi-Square Distribution

x = np.linspace(0, 30, 200)

fig, ax = plt.subplots(figsize=(10, 6))

for df in [1, 2, 3, 5, 10]:
    ax.plot(x, stats.chi2(df).pdf(x), linewidth=2, label=f'df={df}')

ax.set_title('Chi-Square Distribution')
ax.set_xlabel('x')
ax.set_ylabel('Density')
ax.legend()
ax.grid(alpha=0.3)
ax.set_ylim(0, 0.5)
plt.show()

---

Discrete Distributions

1. Binomial Distribution

n = 20
x = np.arange(0, n + 1)

fig, ax = plt.subplots(figsize=(10, 6))

for p in [0.2, 0.5, 0.7]:
    pmf = stats.binom(n, p).pmf(x)
    ax.bar(x + (p - 0.5) * 0.25, pmf, width=0.25, alpha=0.7, label=f'p={p}')

ax.set_title(f'Binomial Distribution (n={n})')
ax.set_xlabel('k')
ax.set_ylabel('P(X = k)')
ax.legend()
ax.grid(alpha=0.3, axis='y')
plt.show()

2. Poisson Distribution

x = np.arange(0, 20)

fig, ax = plt.subplots(figsize=(10, 6))

for lam in [1, 4, 10]:
    pmf = stats.poisson(lam).pmf(x)
    ax.plot(x, pmf, 'o-', linewidth=2, markersize=6, label=f'λ={lam}')

ax.set_title('Poisson Distribution')
ax.set_xlabel('k')
ax.set_ylabel('P(X = k)')
ax.legend()
ax.grid(alpha=0.3)
plt.show()

3. Geometric Distribution

x = np.arange(1, 15)

fig, ax = plt.subplots(figsize=(10, 6))

for p in [0.2, 0.5, 0.8]:
    pmf = stats.geom(p).pmf(x)
    ax.bar(x + (p - 0.5) * 0.25, pmf, width=0.25, alpha=0.7, label=f'p={p}')

ax.set_title('Geometric Distribution')
ax.set_xlabel('k (number of trials)')
ax.set_ylabel('P(X = k)')
ax.legend()
ax.grid(alpha=0.3, axis='y')
plt.show()

---

Distribution Comparisons

1. Normal vs t-Distribution

x = np.linspace(-5, 5, 200)

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# PDF comparison
axes[0].plot(x, stats.norm.pdf(x), 'b-', linewidth=2, label='Normal')
axes[0].plot(x, stats.t(5).pdf(x), 'r-', linewidth=2, label='t (df=5)')
axes[0].fill_between(x, stats.norm.pdf(x), stats.t(5).pdf(x), alpha=0.3)
axes[0].set_title('PDF Comparison')
axes[0].legend()
axes[0].grid(alpha=0.3)

# Tail comparison
x_tail = np.linspace(2, 5, 100)
axes[1].plot(x_tail, stats.norm.pdf(x_tail), 'b-', linewidth=2, label='Normal')
axes[1].plot(x_tail, stats.t(5).pdf(x_tail), 'r-', linewidth=2, label='t (df=5)')
axes[1].set_title('Right Tail Comparison')
axes[1].legend()
axes[1].grid(alpha=0.3)

plt.suptitle('Normal vs t-Distribution', fontsize=14, fontweight='bold')
plt.tight_layout()
plt.show()

2. Exponential Family

x = np.linspace(0, 10, 200)

fig, ax = plt.subplots(figsize=(10, 6))

# Exponential
ax.plot(x, stats.expon(scale=2).pdf(x), label='Exponential (λ=0.5)')

# Gamma
ax.plot(x, stats.gamma(a=2, scale=1).pdf(x), label='Gamma (k=2, θ=1)')

# Chi-square
ax.plot(x, stats.chi2(4).pdf(x), label='Chi-square (df=4)')

ax.set_title('Exponential Family Distributions')
ax.set_xlabel('x')
ax.set_ylabel('Density')
ax.legend()
ax.grid(alpha=0.3)
plt.show()

---

Bivariate Distributions

1. Bivariate Normal

from scipy import stats

x = np.linspace(-3, 3, 100)
y = np.linspace(-3, 3, 100)
X, Y = np.meshgrid(x, y)
pos = np.dstack((X, Y))

fig, axes = plt.subplots(1, 3, figsize=(15, 4))

rhos = [-0.7, 0, 0.7]

for ax, rho in zip(axes, rhos):
    rv = stats.multivariate_normal([0, 0], [[1, rho], [rho, 1]])
    Z = rv.pdf(pos)
    cf = ax.contourf(X, Y, Z, levels=15, cmap='Blues')
    ax.contour(X, Y, Z, levels=8, colors='navy', linewidths=0.5)
    ax.set_title(f'ρ = {rho}')
    ax.set_aspect('equal')
    plt.colorbar(cf, ax=ax)

plt.suptitle('Bivariate Normal Distribution', fontsize=14, fontweight='bold')
plt.tight_layout()
plt.show()

2. Bivariate Normal with Marginals

from mpl_toolkits.axes_grid1 import make_axes_locatable

np.random.seed(42)
rho = 0.6
mean = [0, 0]
cov = [[1, rho], [rho, 1]]
data = np.random.multivariate_normal(mean, cov, 500)

fig, ax_main = plt.subplots(figsize=(8, 8))
divider = make_axes_locatable(ax_main)
ax_top = divider.append_axes("top", 1.2, pad=0.1, sharex=ax_main)
ax_right = divider.append_axes("right", 1.2, pad=0.1, sharey=ax_main)

# Main scatter
ax_main.scatter(data[:, 0], data[:, 1], alpha=0.5, s=20)
ax_main.set_xlabel('X')
ax_main.set_ylabel('Y')

# Marginals
ax_top.hist(data[:, 0], bins=30, density=True, alpha=0.7)
x_range = np.linspace(-4, 4, 100)
ax_top.plot(x_range, stats.norm.pdf(x_range), 'r-', linewidth=2)
plt.setp(ax_top.get_xticklabels(), visible=False)

ax_right.hist(data[:, 1], bins=30, density=True, alpha=0.7, orientation='horizontal')
ax_right.plot(stats.norm.pdf(x_range), x_range, 'r-', linewidth=2)
plt.setp(ax_right.get_yticklabels(), visible=False)

plt.suptitle(f'Bivariate Normal (ρ={rho}) with Marginals', fontsize=13, y=1.02)
plt.show()

---

Distribution Gallery

Complete Overview

fig, axes = plt.subplots(3, 3, figsize=(15, 12))

# Normal
x = np.linspace(-4, 4, 200)
axes[0, 0].plot(x, stats.norm.pdf(x), 'b-', linewidth=2)
axes[0, 0].fill_between(x, stats.norm.pdf(x), alpha=0.3)
axes[0, 0].set_title('Normal')

# Exponential
x = np.linspace(0, 6, 200)
axes[0, 1].plot(x, stats.expon.pdf(x), 'g-', linewidth=2)
axes[0, 1].fill_between(x, stats.expon.pdf(x), alpha=0.3)
axes[0, 1].set_title('Exponential')

# Uniform
x = np.linspace(-0.5, 1.5, 200)
axes[0, 2].plot(x, stats.uniform.pdf(x), 'r-', linewidth=2)
axes[0, 2].fill_between(x, stats.uniform.pdf(x), alpha=0.3)
axes[0, 2].set_title('Uniform')

# Gamma
x = np.linspace(0, 15, 200)
axes[1, 0].plot(x, stats.gamma(a=3).pdf(x), 'purple', linewidth=2)
axes[1, 0].fill_between(x, stats.gamma(a=3).pdf(x), alpha=0.3, color='purple')
axes[1, 0].set_title('Gamma (k=3)')

# Beta
x = np.linspace(0, 1, 200)
axes[1, 1].plot(x, stats.beta(2, 5).pdf(x), 'orange', linewidth=2)
axes[1, 1].fill_between(x, stats.beta(2, 5).pdf(x), alpha=0.3, color='orange')
axes[1, 1].set_title('Beta (α=2, β=5)')

# Chi-square
x = np.linspace(0, 20, 200)
axes[1, 2].plot(x, stats.chi2(5).pdf(x), 'brown', linewidth=2)
axes[1, 2].fill_between(x, stats.chi2(5).pdf(x), alpha=0.3, color='brown')
axes[1, 2].set_title('Chi-square (df=5)')

# Binomial
x = np.arange(0, 21)
axes[2, 0].bar(x, stats.binom(20, 0.5).pmf(x), color='steelblue', alpha=0.7)
axes[2, 0].set_title('Binomial (n=20, p=0.5)')

# Poisson
x = np.arange(0, 15)
axes[2, 1].bar(x, stats.poisson(5).pmf(x), color='seagreen', alpha=0.7)
axes[2, 1].set_title('Poisson (λ=5)')

# Geometric
x = np.arange(1, 12)
axes[2, 2].bar(x, stats.geom(0.3).pmf(x), color='coral', alpha=0.7)
axes[2, 2].set_title('Geometric (p=0.3)')

for ax in axes.flat:
    ax.grid(alpha=0.3)

plt.suptitle('Common Probability Distributions', fontsize=14, fontweight='bold')
plt.tight_layout()
plt.show()

---

Publication-Quality Figure

fig, axes = plt.subplots(2, 2, figsize=(12, 10))

# Normal with shaded regions
x = np.linspace(-4, 4, 200)
rv = stats.norm()
axes[0, 0].plot(x, rv.pdf(x), 'steelblue', linewidth=2)
axes[0, 0].fill_between(x, rv.pdf(x), where=(x >= -1) & (x <= 1), alpha=0.4, color='steelblue')
axes[0, 0].fill_between(x, rv.pdf(x), where=(x >= -2) & (x <= 2), alpha=0.2, color='steelblue')
axes[0, 0].axvline(-1, color='gray', linestyle=':', alpha=0.7)
axes[0, 0].axvline(1, color='gray', linestyle=':', alpha=0.7)
axes[0, 0].set_title('Standard Normal with σ Regions', fontsize=12)
axes[0, 0].set_xlabel('$x$')
axes[0, 0].set_ylabel('$f(x)$')

# t-distribution comparison
axes[0, 1].plot(x, rv.pdf(x), 'b-', linewidth=2, label='Normal')
for df, color in [(3, 'orange'), (10, 'green')]:
    axes[0, 1].plot(x, stats.t(df).pdf(x), color=color, linewidth=2, label=f't (df={df})')
axes[0, 1].set_title('Normal vs t-Distribution', fontsize=12)
axes[0, 1].legend()
axes[0, 1].set_xlabel('$x$')
axes[0, 1].set_ylabel('$f(x)$')

# Gamma family
x = np.linspace(0, 15, 200)
for k, color in [(1, 'red'), (2, 'green'), (5, 'blue')]:
    axes[1, 0].plot(x, stats.gamma(a=k).pdf(x), color=color, linewidth=2, label=f'k={k}')
axes[1, 0].set_title('Gamma Distribution Family', fontsize=12)
axes[1, 0].legend()
axes[1, 0].set_xlabel('$x$')
axes[1, 0].set_ylabel('$f(x)$')

# Beta distribution
x = np.linspace(0, 1, 200)
params = [(2, 2), (2, 5), (5, 2)]
colors = ['blue', 'green', 'red']
for (a, b), color in zip(params, colors):
    axes[1, 1].plot(x, stats.beta(a, b).pdf(x), color=color, linewidth=2, label=f'α={a}, β={b}')
axes[1, 1].set_title('Beta Distribution', fontsize=12)
axes[1, 1].legend()
axes[1, 1].set_xlabel('$x$')
axes[1, 1].set_ylabel('$f(x)$')

for ax in axes.flat:
    ax.grid(alpha=0.3)
    ax.tick_params(labelsize=10)

plt.suptitle('Probability Distribution Examples', fontsize=14, fontweight='bold')
plt.tight_layout()
plt.show()

---

Summary Table

Distribution	scipy.stats	Parameters	Support
Normal	norm(loc, scale)	μ, σ	(-∞, ∞)
Exponential	expon(scale=1/λ)	λ	[0, ∞)
Gamma	gamma(a, scale)	k, θ	[0, ∞)
Beta	beta(a, b)	α, β	[0, 1]
t	t(df)	df	(-∞, ∞)
Chi-square	chi2(df)	df	[0, ∞)
Binomial	binom(n, p)	n, p	{0,...,n}
Poisson	poisson(mu)	λ	{0,1,2,...}
Geometric	geom(p)	p	{1,2,3,...}