Model events over an interval

For X~Poisson(λ), P(X=x)=exp(-λ)λ^x/x! for x=0,1,... and λ>0. Both mean and variance equal λ. A constant-rate Poisson process motivates the model when disjoint intervals have independent counts and multiple events in a tiny interval are negligible. Change the mean in proportion to exposure: doubling time doubles λ, assuming constant rate.