PoissonRate

PoissonRate is the rate parameter, typically denoted by λ, that governs the Poisson distribution and the Poisson process. It represents the average number of events that occur per unit time (or per unit space) under the model.

In a homogeneous Poisson process with rate λ, the count N(t) of events in a time interval of length t follows a Poisson distribution with mean and variance λt. The probability of observing k events in that interval is P(N(t) = k) = e^{-λt} (λt)^k / k!. In such a process, increments over disjoint intervals are independent, and inter-arrival times are exponentially distributed with rate λ (mean 1/λ).

In an inhomogeneous (or non-homogeneous) Poisson process, the rate is a function of time, λ(t). The expected number of events in [0, T] is Λ(T) = ∫_0^T λ(u) du. The stochastic properties depend on this cumulative rate Λ. Estimation follows different paths: for a homogeneous rate, the maximum likelihood estimate is λ̂ = N / T; for non-homogeneous cases, methods include nonparametric kernel estimation or parametric models of λ(t).

Applications of PoissonRate arise across disciplines, including modeling customer arrivals, photon counts in physics, network packet events, and neuronal spiking. The concept also underpins reliability and queuing analyses, where arrival patterns affect system performance.

Limitations should be noted: the Poisson model stipulates independence and equal mean and variance, which may be violated in real data. When events cluster or exhibit overdispersion, alternatives such as the negative binomial model, Hawkes processes, or time-varying-rate models may be more appropriate.