Foundational Statistical Mechanics

Common Probability Distributions

Interactive visual catalog of key continuous and discrete distributions in data science

Probability Density vs Mass

A probability distribution defines how likelihood is allocated over the possible outcomes of a random variable $X$. Continuous variables possess a Probability Density Function (PDF) $f(x)$ where probabilities correspond to integrals over intervals:

$$P(a \le X \le b) = \int_a^b f(x)\,dx, \quad F(x) = P(X \le x) = \int_{-\infty}^x f(t)\,dt$$

Discrete variables possess a Probability Mass Function (PMF) $p(k) = P(X = k)$ where probabilities sum directly: $\sum_k p(k) = 1$.

Distribution Workbench

Select Distribution

Real-World Scenario Preset

Lower Bound ($x_1$): -1.96

Upper Bound ($x_2$): 1.96

Density / Mass Function (PDF / PMF) Continuous

Cumulative Distribution Function (CDF) P(X ≤ x)

Interval Probability P(x₁ ≤ X ≤ x₂) 0.9500

Expected Value E[X] (Mean) 0.000

Variance Var(X) 1.000

Standard Deviation (σ) 1.000

Mathematical Formula

Real-World Applications in Data Science

Distribution Decision Matrix

Distribution	Type	Support	Primary Use Cases in Practice
Normal (Gaussian)	Continuous	$(-\infty, +\infty)$	Central Limit Theorem, linear model residuals, baseline statistical inference ($z$-tests).
Student's $t$	Continuous	$(-\infty, +\infty)$	Small sample mean inference when population variance $\sigma$ is unknown ($t$-tests).
Binomial	Discrete	$\{0, 1, \dots, n\}$	A/B testing click-through rates, binary conversion tracking, coin flips.
Poisson	Discrete	$\{0, 1, 2, \dots\}$	Call arrivals, server request rates, rare defect counting per unit interval.
Exponential	Continuous	$[0, +\infty)$	Time between events, customer churn duration, survival analysis, memoryless decay.
Chi-Square ($\chi^2$)	Continuous	$[0, +\infty)$	Goodness-of-fit tests, contingency table independence, sample variance distribution.
Log-Normal	Continuous	$(0, +\infty)$	Network latency, personal income / financial wealth, file size distributions.