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Determination of the Photon Count Rate, λ Let Q(N, t, λ) dλ = the probability that, given N counts in time interval t, the true count rate is between λ and λ + dλ. We expect a priori that Q(N, t, λ) PN ( λ, t), the probability that a count rate of λ, observed over a time interval t produces N counts. Hence we can write Q(N, t, λ) = kPN ( λ, t). But e t ( t ) N PN ( , t ) , N! and normalization requires that 1 Q ( N , t , )d 0 k t k x N k N e ( t ) d e x dx . 0 0 N! tN! t So k = t and therefore Q( N , t , ) te t (t ) N tPN ( , t). N! From this result we can infer the mean, or expectation value for λ: 0 0 Q ( N , t , )d e t (t ) N 1 d 1 N! N !t For large N N 1 N . t t 0 e x x N 1dx ( N 1)! N 1 . N !t t