Compound Poisson Counts
Add one- and two-unit jumps to Poisson arrivals and see why count variance can exceed the mean.
Free theory articleAdd one- and two-unit jumps to Poisson arrivals and see why count variance can exceed the mean.
Concept & interpretation
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.
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.