Five Distributions Every Simulation Needs

PDF and CDF: two views of the same distribution

Every continuous distribution has a probability density function (PDF), whose height at a point is not itself a probability but a density — only the area under the curve between two points gives you the probability of landing in that range. Its running total, the cumulative distribution function (CDF), is the integral of the PDF up to a point and directly answers "what's the probability of a value at or below x" — always starting at 0, always ending at 1, always non-decreasing. Discrete distributions replace the PDF with a probability mass function, but the same CDF idea applies: it's simply the running sum instead of a running integral.

live demo · distribution shapes and live statistics● LIVE

Normal, Uniform: the two everyone starts with

The Uniform distribution is the simplest possible case: every value in a range is equally likely, giving a flat PDF and a perfectly straight-line CDF. The Normal (Gaussian) distribution is the famous bell curve, defined entirely by its mean μ and standard deviation σ, and it shows up constantly because of the Central Limit Theorem: sum or average enough independent random quantities, regardless of their own individual distribution, and the result converges toward a Normal distribution. That's why measurement errors, human heights and exam scores all cluster into roughly bell-shaped curves — they're each the aggregate of many small, mostly independent contributing factors.

Normal PDF:   f(x) = 1/(σ√(2π)) · exp( −(x−μ)² / (2σ²) )
Uniform PDF:  f(x) = 1/(b−a)     for a ≤ x ≤ b,  else 0

Binomial and Poisson: counting successes and rare events

The Binomial distribution counts the number of successes in a fixed number of independent yes/no trials, each with the same success probability p — coin flips, A/B test conversions, pass/fail quality checks. The Poisson distribution instead counts how many times a rare, independent event happens over a fixed interval of time or space, and it emerges mathematically as the limit of the Binomial distribution when you cut that interval into an enormous number of tiny sub-intervals, each with a vanishingly small chance of one event — which is exactly the structure behind radioactive decay counts, call-centre arrivals, and typos per page.

Binomial PMF: P(X=k) = C(n,k) · p^k · (1−p)^(n−k)     k successes in n trials
Poisson PMF:  P(X=k) = (λ^k · e^(−λ)) / k!             λ = average rate

Exponential: the memoryless waiting-time distribution

If events happen as a Poisson process at a steady average rate, the Exponential distribution describes the waiting time between one event and the next — the time until the next radioactive decay, the next customer, the next server failure. It has an unusual and genuinely useful property called memorylessness: the probability of waiting another 5 minutes for the next event is exactly the same whether you've already waited 0 minutes or 50, because a truly random, constant-rate process carries no memory of how long it's already been running. This is why the Exponential distribution is the default choice for modelling component lifetimes under random (not wear-out) failure and inter-arrival times in queueing models.

Reading two distributions side by side

Overlaying two distributions on the same axes is the fastest way to build intuition for parameters most people otherwise only see as symbols. Two Normal curves with the same mean but different σ show how spread — not location — controls how tightly values cluster; a Poisson distribution with growing λ visibly stretches and starts to look Normal-shaped, a direct illustration of the Central Limit Theorem at work on a discrete distribution; and an Exponential curve next to a Uniform one makes the qualitative difference between "constant hazard, decaying likelihood of large values" and "every value equally likely" immediately visible rather than abstract.

Frequently asked questions

What is the actual difference between a PDF and a CDF?

A probability density function (PDF) describes how likely values are relative to each other — its height at a point is a density, not a probability, and only areas under it correspond to actual probabilities. A cumulative distribution function (CDF) is the running total of that density up to a given value, so it directly reads off as P(X is less than or equal to x), always rises from 0 to 1, and is simply the integral of the PDF.

Why does the Poisson distribution show up so often for rare, random events?

The Poisson distribution is the limiting case of the Binomial distribution when you split a fixed time or space interval into an enormous number of tiny sub-intervals, each with a tiny, independent probability of containing one event. That's exactly the structure of many real processes — radioactive decays, customer arrivals, typos per page — which is why the Poisson distribution keeps appearing for counts of rare, independent events over a fixed interval.

How is the Exponential distribution related to the Poisson distribution?

They describe the same underlying random process from two different angles. If events occur as a Poisson process at a constant average rate, the number of events in a fixed time window follows a Poisson distribution, while the waiting time between consecutive events follows an Exponential distribution with the reciprocal rate parameter — one counts events, the other times the gaps between them.

Try it live

Everything above runs in your browser — open Probability Distributions Explorer and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Probability Distributions Explorer simulation

Five Distributions Every Simulation Needs: Normal, Binomial, Poisson, Exponential, Uniform