Poisson Distribution Moments Mean and Variance Calculator

Advanced tool to explore the Poisson model for event counts calculate mean and variance plus skewness and kurtosis enter the rate per interval choose units inspect stepwise derivations export tables check parameter bounds test sample datasets learn assumptions and apply results in reliability queuing epidemiology and operations research quality control forecasting inventory science education

Inputs

λ must be > 0
How to Use
  1. Choose Direct λ or Rate × Interval.
  2. Enter the parameter(s). λ must be > 0.
  3. Optionally specify k for PMF at k and t for MGF at t.
  4. Press Calculate Moments to view results and formulas.

Tip: When λ is small, skewness and excess kurtosis are large. As λ grows, the distribution approaches normality.

Frequently Asked Questions

λ is the expected number of events in the chosen interval. It equals rate × interval if you provide those inputs.
For the Poisson model, the mean and variance are both λ. This is a defining property that often helps diagnose model fit.
Central moments are taken about the mean (e.g., variance). Raw moments are taken about zero. Both sets are provided up to the fourth order.
Excess kurtosis measures tail heaviness relative to the normal distribution. For Poisson it equals 1/λ.
Yes, for sufficiently large λ the Poisson distribution is well approximated by Normal(λ, λ). Rules of thumb often suggest λ ≳ 10 for reasonable accuracy.
The moment generating function is M(t) = exp(λ(e^t − 1)). Differentiating at t = 0 yields raw moments. You can also evaluate M(t) at a specific t in this tool.
It provides the probability of observing exactly k events. This is useful for predictions and goodness-of-fit checks for discrete counts.
If variance substantially exceeds the mean, consider alternative models like Negative Binomial or Poisson mixtures. Overdispersion suggests the simple Poisson may not be adequate.

Educational use only. Verify results before deployment in critical applications.

Related Calculators

Uniform Distribution PDF/PMF Value Calculator
Pareto Distribution Quantile Percent Point Calculator
Cauchy Distribution CDF Value Calculator
Laplace Distribution CDF Value Calculator
Geometric Distribution Moments Mean Variance Calculator
Negative Binomial Distribution PDF/PMF Value Calculator
Square Area Calculator
Rectangle Perimeter Calculator
Rectangle Circumradius Calculator

Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.