Poisson laws and rare-event approximations

Try

An original synthetic model assigns a mean of two events to a fixed observation interval. A Poisson count permits zero, one, two and arbitrarily many events, rather than imposing a fixed maximum. Predict whether knowing this one interval’s count distribution determines how counts in neighboring intervals depend on each other. It does not specify their joint law.

Predict

Compare P(X=2) and P(X=1) when λ=2. The common exponential factor will cancel in their ratio. Then predict how increasing interval duration affects the parameter if a constant-rate Poisson process is explicitly assumed. The rate and the interval mean are related but have different units.

Experiment

For X~Poisson(2), the masses are e^(−2)2^k/k! for k=0,1,2,… . The first three are e^(−2), 2e^(−2), 2e^(−2). Thus the ratio P(X=2)/P(X=1)=1. Build further masses recursively using P(X=k+1)/P(X=k)=2/(k+1), which avoids recalculating factorials.

If a process has rate two events per hour with independent stationary increments, a three-hour count is Poisson(6). The expected count is six, not a guarantee. Write both the rate unit and the dimensionless interval parameter next to the model.

Understand

The Poisson parameter λ is positive, and the mass function is e^(−λ)λ^k/k!. Normalization follows from the exponential series: e^(−λ)Σλ^k/k!=1. Its mean and variance both equal λ. This equality is a model property, not a universal feature of count data; overdispersion can indicate heterogeneity or dependence.

For the mean, multiply the mass by k and shift the index. For the factorial second moment, k(k−1) similarly gives E[X(X−1)]=λ². Hence E(X²)=λ²+λ and variance is λ. These derivations connect series manipulation to moments rather than treating the formulas as unrelated facts.

Independent Poisson variables add to another Poisson variable with parameter equal to the sum. Independence is required for that distributional conclusion. Linearity of expectation still adds their means without independence, but their variance and full sum law may differ. A Poisson marginal for one interval does not establish independent increments across time.

The Poisson process is a stronger model specifying a family of counts with a constant rate and independent increments, along with appropriate small-interval behavior. This lesson uses its interval interpretation only when those assumptions are explicitly stated. Real arrivals can cluster, depend on time of day or influence each other, so a process model requires checking beyond a mean count.

The rare-event limit starts with Binomial(n,λ/n) for fixed λ and n large enough that λ/n≤1. For each fixed k, separate the binomial probability into a falling-factorial ratio tending to one, λ^k/k!, a survival factor tending to e^(−λ), and a final factor tending to one. The resulting mass is Poisson(λ). This is a limiting argument, not an assertion of equality for every finite n.

Approximation quality depends on the event probability and intended tail. For a finite application, label the approximation and compare with the binomial calculation where feasible. A shared rare-event story does not override dependence among trials.

Apply

Derive normalization and the mean with explicit index shifts. Then explain why summing two marginally Poisson counts requires a dependence statement before identifying a Poisson sum. Write a count-data pattern that would challenge a constant-rate independent-increment interpretation.

Continue

Next replace masses at integers with a density over a continuous support. Normalization remains essential, but point probability is no longer density height.