The AstroStat Slog

Astrophysics, especially high-energy astrophysics, is all about counting photons. And this, it is said, naturally leads to all our data being generated by a Poisson process. True enough, but most astronomers don’t know exactly how it works out, so this derivation is for them.

Suppose N counts are randomly placed in an interval of duration τ without any preference for appearing in any particular portion of τ. i.e., the distribution is uniform. The counting rate R = N/τ. We can now ask, what is the probability of finding k counts in an infinitesimal interval δt within τ?

First, consider the probability that one count, placed randomly, will fall inside δt,

ρ = δt/τ ≡ Rδt/N ≡ ν/N

where ν = R δt represents the expected counts intensity in the interval δt. When N counts are scattered over τ, the probability that k of them will fall inside δt is described with a binomial distribution,

p(k|ρ,N) = ^NC_k ρ^k (1-ρ)^N-k

as the product of the probability of finding k events inside δt and the probability of finding the remaining events outside, summed over all the possible distinct ways that k events can be chosen out of N. Expanding the expression and rearranging,

= N!/{(N-k)!k!} (R δt/N)^k (1-(R δt/N))^N-k

= N!/{(N-k)!k!} (ν^k/N^k) (1-(ν/N))^N-k

= N!/{(N-k)!N^k} (ν^k/k!) (1-(ν/N))^N (1-(ν/N))^-k

Note that as N,τ —> ∞ (while keeping R fixed),

N!/{(N-k)!N^k} , (1-(ν/N))^-k —> 1
(1-(ν/N))^N —> e^-ν

and the expression reduces to

p(k|ν) = (ν^k/k!) e^-ν

which is the familiar (in a manner of speaking) expression for the Poisson likelihood.