Compound Poisson Counts

Concept Overview

A compound Poisson count represents the total sum of independent, identically distributed jumps that occur according to a Poisson arrival process. If standard arrivals occur at rate λ and each arrival contributes a random jump size instead of a constant 1, the total count over a given time interval is a compound Poisson random variable. This structure models physical and statistical scenarios such as batch arrivals, insurance claims, or variable-sized data packets, explaining why real-world counts often exhibit greater variance than a simple Poisson distribution predicts.

Model Definition

Suppose the number of arrivals is N, where N ~ Poisson(λ). Let J₁, J₂, ..., J_N be independent jump sizes with a known discrete distribution. The total sum S is defined as:

S = J₁ + J₂ + ... + J_N

If each jump is 1 with probability 1-p and 2 with probability p, the expected jump size E[J] is (1-p)×1 + p×2 = 1 + p. By conditioning on N, the expected value and variance of S are:

E[S] = E[N]E[J] = λ(1 + p)

Var[S] = E[N]Var[J] + Var[N](E[J])² = λ E[J²]

Since E[J²] = (1-p)×1² + p×2² = 1 + 3p:
Var[S] = λ(1 + 3p)

Worked Example: PMF Recurrence

Because the jumps can only take values of 1 or 2, the total count S can also be expressed as S = N₁ + 2N₂, where N₁ ~ Poisson(λ(1-p)) and N₂ ~ Poisson(λp) are independent Poisson variables for one-unit and two-unit jumps respectively.

Let λ₁ = λ(1-p) and λ₂ = λp. The exact finite probability mass function (PMF) f_n = P(S = n) satisfies a recurrence derived from the probability generating function:

f₀ = e^-λ
f₁ = λ₁ f₀
n f_n = λ₁ f_n-1 + 2 λ₂ f_n-2 (for n ≥ 2)

Start with λ = 2 and p = 0.5 (two-unit jumps happen half the time).
We have λ₁ = 2(0.5) = 1 and λ₂ = 2(0.5) = 1.
Calculate f₀ = e^-2 ≈ 0.1353.
Calculate f₁ = 1 × f₀ ≈ 0.1353.
For n=2: 2 f₂ = 1 × f₁ + 2(1) f₀ = 3 f₀. Thus f₂ = 1.5 e^-2 ≈ 0.2030.

Experiment: Exploring the Tails

Use the interactive controls at https://riano.app/module/compound-poisson-counts to explore how the compound parameters alter the resulting distribution and its moments.

Set the Poisson mean to 4.
Set the Double-jump quarters to 1 (p = 0.25).
Observe the computed moments: E[S] = 4(1 + 0.25) = 5, and Var[S] = 4(1 + 0.75) = 7. Notice how the variance now exceeds the mean, an indicator of overdispersion due to the compounded jump sizes.
Set the Total probe to 4 to see the specific point mass P(S = 4).
Observe the unplotted tail indicator P(S > 12) to see the probability mass strictly greater than 12.

Limitations

This numerical visualization truncates the PMF vector and the graph view at S = 12. The distribution itself extends to infinity; the unplotted tail probability beyond 12 is calculated explicitly. Also, this particular module only supports jump sizes of exactly 1 or 2. In more general compound Poisson processes, jump sizes could be continuous variables (like Exponential or Lognormal laws) or other discrete distributions.

References

The recurrence identities for the PMF and the moment relations for the compound Poisson process are mathematically detailed in the references below.

Random Services: Compound Poisson Process— Kyle Siegrist's textbook coverage of the definitions, independent increments, and moment formulas (E[S]=λ E[J] and Var[S]=λ E[J²]) for a generic compound Poisson aggregate.

Compound Poisson Counts