Probability distributions

Choose one of the 13 distributions and a query: Value returns density or point mass plus P(X≤x) and P(X>x); Quantile returns the threshold and achieved cumulative probability; Interval returns bounds and achieved coverage. Probabilities are entered/displayed as percentages. Parameters retain their standard meanings: lognormal μ/σ describe log(x), gamma uses shape and rate, geometric counts failures before the first success from zero. Density is not a point probability and can exceed one or diverge at a boundary.

Central, lower and upper intervals use the requested coverage literally, including 0%/100% endpoints. Between uses closed bounds. Discrete quantiles are the smallest support values reaching the requested cumulative probability; achieved interval coverage may exceed the request because of jumps. Noninteger point queries have zero discrete point mass, while their CDF uses the floor. No parameter, probability or integer trial count is silently rounded or clamped.

Plots use a finite window, independent of analytic probability calculations. Broad discrete windows use at most 80 grouped probability bars; selected parts use their actual mass. Unbounded endpoint densities are clipped and marked. Infinities at true support/quantile endpoints are distinguished from numerical failures. Small positive tails use scientific notation; unresolved zeros are marked and rounded full probabilities can show their complement. Iteration/convergence and machine precision limits can prevent reliable evaluation of extreme parameters.

Sample fitting is exploratory, not inference or a goodness-of-fit test. Normal and lognormal use the population second moment (also their plug-in likelihood scale), gamma/beta use population-moment estimates, uniform uses observed extremes, exponential/Poisson/chi-square/geometric use mean plug-ins. Binomial fitting keeps the entered known trial count n and estimates only p; n cannot be inferred from the observed maximum. Invalid or out-of-support data reject the whole fit. Constant data cannot yield nondegenerate normal/lognormal/uniform/gamma/beta fits. Simple fitting is not offered for Student-t, F or Weibull.

Queries update directly as you edit; Enter updates immediately. Fit opens a sample editor and estimates parameters with its button or Ctrl/⌘+Enter in the sample field. One shared sample and number format remain available across families; each family retains its own raw parameter and query drafts, including unfinished and inactive values. Only inputs persist locally through navigation and reload; plots/results are recomputed. Up to 50,000 finite sample values and one million characters. Spaces, lines and semicolons separate values; 1.5 also accepts comma separators, whereas 1,5 uses decimal commas.