Poisson distribution with efficiency problem

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SUMMARY

The discussion focuses on determining the probability distribution function for the number of detector counts, modeled by a Poisson distribution, while incorporating detector efficiency (α). The exponential distribution ε(t,λ) = λexp(-λt) serves as the foundation for this analysis. The participants conclude that if the efficiency is a known constant, it acts as a multiplicative factor, scaling the occurrences and rate. If the efficiency varies with the number of counts (n) and time (t), further investigation is necessary to establish its functional form.

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  • Understanding of Poisson distribution and its properties
  • Familiarity with exponential distribution and its applications
  • Knowledge of statistical efficiency concepts
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atomi
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Hi,

I have a problem with determining the probability distribution function of the number n of detector counts in a given time t. I am assuming the events follow exponential distribution ε(t,λ) = λexp(-λt). Now if that was everything it would simply be a Poisson distribution, however, what I need to acount for is the detector efficiency α (0-1) and I can't think of a way how to do it. Could anyone give me a hint?

cheers
 
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Is the efficiency a known constant? If so, then you just have a multiplicative constant because P(detection/occurrence) = constant. Then the number of occurrences and the corresponding rate are scaled by the efficiency. If it's not a known constant then I would think that it must be determined. If the efficiency is a function of n and t then I would have to think about it.
 

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