Comparing λ Across Groups

The table below shows the probability (as a percentage) that each group’s rate λ is the highest or lowest among all groups (approximated via Monte Carlo simulation).

Group	Prob(Highest λ)	Prob(Lowest λ)

The Poisson Distribution Across Multiple Groups

\[ P(k \mid \lambda, t) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}, \quad \lambda \sim \mathrm{Gamma}(\alpha, \beta). \]

When you have multiple Poisson processes (groups), each may have a different underlying rate \(\lambda_i\). By treating each group’s unknown \(\lambda_i\) as a random variable with a Gamma posterior, you can visualize the PDFs together and compare their likelihoods of being the highest or lowest among the groups.