Photon Counting Statistics
Poisson statistics, SNR, dead time corrections, and count distributions.
SNR per Interval √μ
10.0 (20.0 dB)
Relative Error 1/√μ
10.00%
Dead Time Loss
4.8%
recorded 0.952 of 1.000 Mcps
Corrected-Rate SNR in 1 s
975.9
√N, N = 9.524e+5 recorded
P(n) = μn·e−μ / n! [Poisson]; SNR = √μ (shot-noise limit)
m = R / (1 + R·τ) [non-paralyzable dead time]; corrected R̂ = m / (1 − m·τ)
Recorded counts are sub-Poissonian: Var N / N = 1/(1 + R·τ)² = 0.9070. The correction amplifies that smaller spread back, so the corrected rate has relative error 1/√N of the recorded counts.
Count Distribution
SNR & Error vs Counts
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Exact Poisson counting statistics for an ideal detector, with a non-paralyzable dead time. The corrected-rate SNR √N of the recorded counts holds for long gates, where the recorded counts have the renewal-process variance N/(1 + Rτ)².
- Knoll G. F. (2010). Radiation Detection and Measurement, 4th ed. Wiley. Ch. 4, section VII: non-paralyzable m = n/(1 + nτ) and paralyzable m = n·e^(−nτ) dead-time models.
- Müller J. W. (1973). Dead-time problems. Nucl. Instrum. Methods 112, 47–57. Dead-time models and the statistics of the recorded counts. doi:10.1016/0029-554X(73)90773-8