Poisson and Exponential Distribution Simulation

The interface runs in the browser, while random generation, statistics, distribution calculations, and Excel creation run on the backend server.

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Simulation controls

Meaning of the two distributions

Poisson distribution

Counts the number of events during a fixed interval of length T.

P(N(T) = k) = e−λT(λT)k / k!

Variable type: discrete.

Possible values: 0, 1, 2, 3, …

Exponential distribution

Measures the continuous waiting time between two consecutive events.

f(x) = λe−λx,   x ≥ 0

Variable type: continuous.

Possible values: every non-negative real number.

Real-time animation

Poisson view: events counted in fixed windows

Each vertical block represents one fixed interval of length T.

Exponential view: time between events

Each event creates a new interarrival-time observation.

Simulation time0.00
Current interval count0
Completed Poisson intervals0
Exponential samples0
Mean events per interval0.00
Theoretical λT4.00
Mean interarrival time0.00
Theoretical 1/λ0.50

Empirical and theoretical distributions

Poisson probability mass function

Exponential probability density

Main differences

CharacteristicPoissonExponential
Question answeredHow many events occur during a fixed interval?How long must we wait until the next event?
Random variableNumber of eventsTime between events
Variable typeDiscreteContinuous
Expected valueE[N(T)] = λTE[X] = 1/λ
Typical logistics exampleNumber of trucks arriving at a gate within one hour.Time between consecutive truck arrivals.