Common Probability Distributions and Their Visualization in Python

In this article we introduce a range of common probability distributions and show how to visualize them using Python libraries such as NumPy, SciPy, Matplotlib, and Seaborn.

Uniform Distribution : The uniform distribution can be discrete (e.g., a fair die) or continuous over an interval. The following code generates and plots both forms.

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

# continuous uniform
a = 0
b = 50
size = 5000
X_continuous = np.linspace(a, b, size)
continuous_uniform = stats.uniform(loc=a, scale=b)
continuous_uniform_pdf = continuous_uniform.pdf(X_continuous)

# discrete uniform
X_discrete = np.arange(1, 7)
discrete_uniform = stats.randint(1, 7)
discrete_uniform_pmf = discrete_uniform.pmf(X_discrete)

# plot both
fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(15,5))
ax[0].bar(X_discrete, discrete_uniform_pmf)
ax[0].set_xlabel("X")
ax[0].set_ylabel("Probability")
ax[0].set_title("Discrete Uniform Distribution")
ax[1].plot(X_continuous, continuous_uniform_pdf)
ax[1].set_xlabel("X")
ax[1].set_ylabel("Probability")
ax[1].set_title("Continuous Uniform Distribution")
plt.show()

Normal (Gaussian) Distribution : The most familiar distribution, symmetric around its mean. The code below plots its probability density function.

mu = 0
variance = 1
sigma = np.sqrt(variance)
x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)

plt.subplots(figsize=(8,5))
plt.plot(x, stats.norm.pdf(x, mu, sigma))
plt.title("Normal Distribution")
plt.show()

Lognormal Distribution : A continuous distribution whose logarithm is normally distributed. The script visualizes three parameter settings.

X = np.linspace(0, 6, 500)

std = 1
mean = 0
lognorm_distribution = stats.lognorm([std], loc=mean)
lognorm_distribution_pdf = lognorm_distribution.pdf(X)
fig, ax = plt.subplots(figsize=(8,5))
plt.plot(X, lognorm_distribution_pdf, label="μ=0, σ=1")
ax.set_xticks(np.arange(min(X), max(X)))

std = 0.5
mean = 0
lognorm_distribution = stats.lognorm([std], loc=mean)
lognorm_distribution_pdf = lognorm_distribution.pdf(X)
plt.plot(X, lognorm_distribution_pdf, label="μ=0, σ=0.5")

std = 1.5
mean = 1
lognorm_distribution = stats.lognorm([std], loc=mean)
lognorm_distribution_pdf = lognorm_distribution.pdf(X)
plt.plot(X, lognorm_distribution_pdf, label="μ=1, σ=1.5")

plt.title("Lognormal Distribution")
plt.legend()
plt.show()

Poisson Distribution : Models the count of events occurring in a fixed interval. The first snippet computes a probability mass, and the second visualizes a sample.

from scipy import stats
print(stats.poisson.pmf(k=9, mu=3))

X = stats.poisson.rvs(mu=3, size=500)
plt.subplots(figsize=(8,5))
plt.hist(X, density=True, edgecolor="black")
plt.title("Poisson Distribution")
plt.show()

Exponential Distribution : Describes the time between events in a Poisson process. The code plots its PDF.

X = np.linspace(0, 5, 5000)
exponential_distribution = stats.expon.pdf(X, loc=0, scale=1)
plt.subplots(figsize=(8,5))
plt.plot(X, exponential_distribution)
plt.title("Exponential Distribution")
plt.show()

Binomial Distribution : Represents the number of successes in a fixed number of independent Bernoulli trials. The snippet generates a sample and plots a histogram.

X = np.random.binomial(n=1, p=0.5, size=1000)
plt.subplots(figsize=(8,5))
plt.hist(X)
plt.title("Binomial Distribution")
plt.show()

Student's t Distribution : Used when estimating the mean of a normally distributed population with unknown variance. The following code visualizes t‑distributions with different degrees of freedom.

import seaborn as sns
from scipy import stats

X1 = stats.t.rvs(df=1, size=4)
X2 = stats.t.rvs(df=3, size=4)
X3 = stats.t.rvs(df=9, size=4)

plt.subplots(figsize=(8,5))
sns.kdeplot(X1, label="1 d.o.f")
sns.kdeplot(X2, label="3 d.o.f")
sns.kdeplot(X3, label="6 d.o.f")
plt.title("Student's t distribution")
plt.legend()
plt.show()

Chi‑squared Distribution : A special case of the gamma distribution, often used in hypothesis testing. The code below plots PDFs for 1, 2, and 3 degrees of freedom.

X = np.arange(0, 6, 0.25)
plt.subplots(figsize=(8,5))
plt.plot(X, stats.chi2.pdf(X, df=1), label="1 d.o.f")
plt.plot(X, stats.chi2.pdf(X, df=2), label="2 d.o.f")
plt.plot(X, stats.chi2.pdf(X, df=3), label="3 d.o.f")
plt.title("Chi-squared Distribution")
plt.legend()
plt.show()

Understanding these distributions and being able to visualize them is essential for data science, statistical analysis, and machine‑learning tasks.