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Statistics · Distributions

Erlang Distribution Calculator

Calculate density, cumulative and survival probability, quantiles, and moments for waiting time until an integer-numbered Poisson event.

Statistics · Distributions

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Private in-browser calculation · explicit assumptions
Continuous support is shown as a scale. Probability belongs to area over an interval, not to the density height at one point.
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  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
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Understand the distribution

What the erlang distribution calculator models

Modeling positive waiting times or lifetimes. Positive-support families offer different hazard and shape behavior. Parameter names and rate-versus-scale conventions must match the formula being used.

Density and cumulative rule

F(x)=1−e^(−λx)Σⱼ₌₀^(k−1)(λx)^j/j!, with integer shape k≥1.

A density height is not itself a probability. Continuous probability is the area accumulated across an interval.

Interpretation and limits

Erlang is the gamma special case with integer shape and has a direct waiting-time-to-the-kth-event interpretation.

Model boundary: The underlying event process must have independent exponential gaps with a constant rate; rate and time units must be reciprocal.

Quick guide

How to use this calculator

  1. Enter the observations, probabilities, model parameters, or summary statistics requested by the visible labels.
  2. Keep every value on the same scale and confirm that the selected sampling relationship, distribution, and tail convention match the question you are investigating.
  3. Read the result together with its assumptions and interpretation. Statistical output summarizes uncertainty under a model; it does not repair biased data or establish causation.

Calculation method

How the erlang distribution calculator works

F(x)=1−e^(−λx)Σⱼ₌₀^(k−1)(λx)^j/j!, with integer shape k≥1.

Erlang is the gamma special case with integer shape and has a direct waiting-time-to-the-kth-event interpretation.

Worked example

Erlang Distribution example

Shape 2 and rate 1 give cumulative probability 1−3e⁻² at x=2.

F(x)=1−e^(−λx)Σⱼ₌₀^(k−1)(λx)^j/j!, with integer shape k≥1.

Supported inputs

Precision and limits

Model and design

The underlying event process must have independent exponential gaps with a constant rate; rate and time units must be reciprocal.

Numerical scope

Inputs use double-precision numerical methods with guarded domains. Datasets accept up to 10,000 finite plain-decimal values. Extremely large parameters or probabilities deep in a numerical tail may require specialist statistical software.

Interpretation

Erlang is the gamma special case with integer shape and has a direct waiting-time-to-the-kth-event interpretation.

Decision boundary

The calculator does not validate how data were collected, diagnose dependence or bias, choose a scientifically meaningful effect, or replace review by a qualified statistician for consequential research, medical, regulatory, safety, or policy decisions.

Privacy

Entered values and calculated results stay in this browser and are not sent to an analytics service.

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