This tool plots seven core probability distributions — Normal, Poisson, Exponential, Binomial, Beta, Uniform and Chi-squared — in either probability-density (PDF) or cumulative (CDF) view. Continuous densities are drawn as smooth curves over a 600-point grid, while discrete distributions appear as bars. Each is computed from its exact closed-form formula, with special functions such as the log-gamma (Lanczos approximation), the error function and regularised incomplete gamma and beta functions handling the heavier mathematics.
The distribution tabs and parameter sliders let you set values such as the Normal mean and standard deviation, the Poisson and Exponential rate, the Binomial trials and success probability, the Beta shape parameters, the Uniform bounds and the Chi-squared degrees of freedom. The interval inputs shade and compute P(a < X < b), the panel reports mean, variance, skewness, kurtosis and quartiles, and the overlay compares two distributions at once. These distributions underpin statistics, risk modelling, quality control and machine learning.
What is a probability distribution?
A probability distribution describes how likely each possible value of a random variable is. For continuous variables this is captured by a probability density function (PDF), whose area under the curve gives probabilities; for discrete variables it is a probability mass function assigning a probability to each outcome. The explorer lets you view and compare seven of the most widely used distributions.
What is the difference between the PDF and CDF views?
The PDF (or PMF for discrete cases) shows the density or probability at each value, so its peak indicates the most likely region. The CDF, F(x), shows the accumulated probability that the variable is at most x, rising from 0 to 1. Toggle the PDF and CDF buttons to switch between them; the CDF view always uses a fixed vertical scale from 0 to 1.
How does the P(a < X < b) calculation work?
Enter values for a and b and the tool shades that region of the PDF and reports the probability. It computes the result exactly as F(b) − F(a) using each distribution's cumulative function, so for the Normal it relies on the error function, for Beta and Chi-squared on regularised incomplete functions, and for discrete distributions it sums the relevant masses.
It covers the Normal N(μ, σ), Poisson(λ), Exponential(λ), Binomial(n, p), Beta(α, β), Uniform on [a, b] and Chi-squared with k degrees of freedom. Three are discrete or continuous as appropriate — Poisson and Binomial are discrete and drawn as bars, while the others are continuous curves.
Skewness measures asymmetry: zero for symmetric distributions like the Normal and Uniform, positive when a long tail stretches to the right, as with the Exponential or low-degree Chi-squared. Excess kurtosis measures tail heaviness relative to the Normal, which has excess kurtosis of zero; negative values, such as the Uniform's −1.2, indicate lighter tails and a flatter shape.
Both are discrete distributions defined only at integer values, so a continuous curve would be misleading. The explorer draws a bar at each integer k whose height equals P(X = k). The Binomial counts successes in n independent trials, while the Poisson counts events in a fixed interval at average rate λ; both are plotted across their full integer range.
The mathematics uses standard high-accuracy methods: a Lanczos approximation for the log-gamma function, a rational approximation for the error function, and continued-fraction or series expansions for the regularised incomplete beta and gamma functions. Quantiles for distributions without closed forms are found by binary search on the CDF, giving results accurate to many decimal places for typical parameter ranges.
They describe the same underlying process from different angles. If events occur randomly at a constant average rate λ, the number of events in a fixed interval follows a Poisson distribution, while the waiting time between consecutive events follows an Exponential distribution. The Exponential is also memoryless, meaning the chance of waiting a further t is unaffected by how long you have already waited.
The Beta distribution is supported on the interval [0, 1], making it a natural model for an unknown probability or proportion. It is the conjugate prior for the Bernoulli and Binomial likelihoods, so updating a Beta prior with observed successes and failures yields another Beta distribution. Setting both shape parameters to 1 recovers the Uniform distribution as a special case.
The Normal models measurement error and aggregated quantities via the Central Limit Theorem; the Poisson models call-centre arrivals or rare-event counts; the Exponential models time between failures; the Binomial models polling and clinical-trial outcomes; the Chi-squared underlies goodness-of-fit and independence tests; and the Beta and Uniform appear throughout Bayesian inference, simulation and random-number generation.