Common Distributions: An Overview

A handful of probability distributions describe a huge range of natural, epidemiological, and experimental data: coin-flip-like proportions, counts of rare events, waiting times between infections, and the bell-shaped noise that emerges whenever many small effects add up. This page is a map to the five workhorse distributions and when each one arises.

The five workhorses

Normal — continuous, symmetric bell curve. Arises whenever many small independent effects add up (heights, measurement error, sample means via the central limit theorem).
Binomial — discrete count of “successes” in $n$ independent yes/no trials. Arises in proportion testing (vaccinated vs. not, cured vs. not).
Poisson — discrete count of rare events in a fixed window. Arises for case counts, mutations, or arrivals when events are independent and the rate is constant.
Exponential — continuous waiting time until the next event. Arises for time between infections or the length of an infectious period; it is the memoryless partner of the Poisson.
Student’s t — continuous, bell-shaped but heavier-tailed than the normal. Arises for the standardized sample mean when the population $\sigma$ is estimated from data.

Comparison table

Distribution	Type	Support	Parameters	Mean	Variance	R sampler
Normal	Continuous	$x\in(-\infty,\infty)$	$\mu,\ \sigma^2$	$\mu$	$\sigma^2$	`rnorm(n, mean, sd)`
Binomial	Discrete	$k\in\{0,1,\dots,n\}$	$n,\ p$	$np$	$np(1-p)$	`rbinom(N, n, p)`
Poisson	Discrete	$k\in\{0,1,2,\dots\}$	$\lambda>0$	$\lambda$	$\lambda$	`rpois(n, lambda)`
Exponential	Continuous	$x\ge 0$	rate $\lambda>0$	$1/\lambda$	$1/\lambda^2$	`rexp(n, rate)`
Student’s t	Continuous	$x\in(-\infty,\infty)$	df $\nu>0$	$0$ (for $\nu>1$ )	$\frac{\nu}{\nu-2}$ (for $\nu>2$ )	`rt(n, df)`

The R d/p/q/r naming convention

R names distribution functions by a prefix + root. The root is the distribution (norm, binom, pois, exp, t), and the prefix picks the operation:

d* — density (pdf) or probability mass (pmf): $f(x)$ .
p* — cumulative probability (cdf): $P(X\le x)$ .
q* — quantile (inverse cdf): the $x$ with $P(X\le x)=p$ .
r* — random sampling: draw variates.

So dnorm, pnorm, qnorm, rnorm are the four faces of the normal distribution, and the same pattern holds for every distribution below.

In code

R

set.seed(42)

# One sampler per distribution (r* prefix)
rnorm(3, mean = 0, sd = 1)   # Normal
rbinom(3, size = 10, prob = 0.3)  # Binomial
rpois(3, lambda = 4)         # Poisson
rexp(3, rate = 0.5)          # Exponential
rt(3, df = 5)                # Student's t

# d/p/q faces of the normal
dnorm(0)          # density at 0  -> 0.3989
pnorm(1.96)       # cdf at 1.96   -> 0.975
qnorm(0.975)      # quantile      -> 1.96

Python

import numpy as np
from scipy import stats

rng = np.random.default_rng(42)

rng.normal(0, 1, size=3)                 # Normal
rng.binomial(n=10, p=0.3, size=3)        # Binomial
rng.poisson(lam=4, size=3)               # Poisson
rng.exponential(scale=1/0.5, size=3)     # Exponential (scale = 1/rate)
stats.t.rvs(df=5, size=3, random_state=42)  # Student's t

stats.norm.pdf(0)      # density  -> 0.3989
stats.norm.cdf(1.96)   # cdf      -> 0.975
stats.norm.ppf(0.975)  # quantile -> 1.96

Julia

using Distributions, Random
Random.seed!(42)

rand(Normal(0, 1), 3)      # Normal
rand(Binomial(10, 0.3), 3) # Binomial
rand(Poisson(4), 3)        # Poisson
rand(Exponential(2), 3)    # Exponential (scale = 1/rate = 2)
rand(TDist(5), 3)          # Student's t

pdf(Normal(0, 1), 0)       # density  -> 0.3989
cdf(Normal(0, 1), 1.96)    # cdf      -> 0.975
quantile(Normal(0, 1), 0.975)  # quantile -> 1.96

Simulation

A quick check that samples converge to their theoretical means. Averaging many Poisson( $\lambda=4$ ) draws should land near $4$ .

set.seed(1)
mean(rpois(1e6, lambda = 4))  # ~ 4.00 (theoretical mean = lambda = 4)
var(rpois(1e6, lambda = 4))   # ~ 4.00 (Poisson: variance = mean)

As the sample size grows the empirical mean and variance both approach $\lambda$ , exactly as the law of large numbers predicts.

Why it matters for statistics

Every inferential method assumes a probability model for the data. Choosing the right distribution — counts vs. proportions vs. continuous measurements — determines which estimator, test, and confidence interval are valid. Recognizing the mechanism that generates each distribution (adding effects → normal, counting rare events → Poisson, waiting for events → exponential) lets you pick models from first principles rather than by trial and error.