Common Probability Distributions and Their Visualization with Python

Probability and statistics are essential foundations for data science and machine learning; understanding data distributions helps model real‑world phenomena and assess variability.

Uniform Distribution

The uniform distribution assigns equal probability to all outcomes. A discrete example is a fair die (1/6 for each face), while a continuous uniform distribution spans a range a to b .

<code>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()</code>

Normal (Gaussian) Distribution

The normal distribution, also known as the Gaussian or bell curve, is defined by its mean μ and standard deviation σ . The mean, median, and mode coincide, and the total area under the curve equals 1.

<code>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()</code>

The empirical rule states that about 68% of data fall within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean.

Log‑Normal Distribution

A log‑normal distribution arises when the logarithm of a variable follows a normal distribution. It is defined only for positive values and produces a right‑skewed curve.

<code>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")
std = 0.5
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()</code>

Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval when events happen at a constant average rate λ . It is a discrete count distribution.

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

<code>0.002700503931560479
</code>

Visualization of a Poisson sample:

<code>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()</code>

Exponential Distribution

The exponential distribution describes the time between successive events in a Poisson process. Its PDF is defined by the rate parameter λ .

<code>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()</code>

Binomial Distribution

The binomial distribution gives the probability of obtaining a fixed number of successes x in n independent Bernoulli trials with success probability p .

<code>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()</code>

Student's t Distribution

The t‑distribution is used when estimating the mean of a normally distributed population with unknown variance, especially for small sample sizes. Its shape approaches the normal distribution as degrees of freedom increase.

<code>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="9 d.o.f")
plt.title("Student's t distribution")
plt.legend()
plt.show()</code>

Chi‑Squared Distribution

The chi‑squared distribution is a special case of the gamma distribution and is widely used for hypothesis testing and confidence interval construction. It is defined by the number of degrees of freedom k .

<code>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()</code>

Mastering these distributions equips data‑science practitioners with the tools needed to model, simulate, and analyze real‑world data effectively.