Birds on a Wire (Poisson Distribution)

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When birds land randomly along a wire, the gaps between neighboring birds follow an exponential distribution — the hallmark of a Poisson process. The average gap equals wire length ÷ number of birds. Adjust the parameters and run the simulation to watch the distribution emerge from randomness. Hover over any histogram bar to highlight the corresponding intervals on the wire. The dashed curve shows the theoretical exponential fit.

Birds

Bird size (m)

Bins

Observed mean — meters

Theoretical mean — meters

Intervals —

Exponential fit P(x) = λe^−λx

What is a Poisson Process?

A Poisson process models events that occur randomly and independently over time or space. Here, each bird chooses its perch position uniformly at random along the wire, with no preference for any location and no coordination with other birds.

The key insight is that the gaps between consecutive events in a Poisson process follow an exponential distribution with mean μ = 1/λ, where λ is the rate (birds per meter). In this simulation, λ = birds ÷ 5 m, so μ = 5 ÷ birds.

Try increasing the number of birds — the observed mean should converge toward the theoretical mean. Set bird size to 0 to see pure point-process behavior, or increase it to see how a minimum spacing (exclusion zone) shifts the distribution rightward.