The count follows Poisson Distribution:

P(n|\lambda) = \frac{(\lambda T)^n}{n!} Exp(-\lambda T)

this means, the probability of number of count, n, happened in time interval T given that the mean rate is \lambda .

now, we measured no count in time interval T, what is the mean rate for given confident interval?

if the count is zero, then the probability is:

P(0|\lambda) = Exp(-\lambda T )

and we can treat this as a new probability density function that, the probability of count rate is \lambda in time interval T.

normalized this pdf.

P(\lambda) = T Exp(-\lambda T )

we can see, for \lambda = 0 , the probability is 1. of course, if the count rate is 0, then no count is 100 %. therefore, we want to estimate the maximum count rate it will be to give “zero count “.

Now we have to introduce the Confident Interval (CL). this is the chance that the “interest” is true. in here, our interest is the “count rate is smaller then some value “. Thus, the total probability is:

1- CL = \int_0^(\lambda_0) P(\lambda) d\lambda

\lambda = \frac{1}{T} ln(\frac{1}{CL})

Lets give an example. suppose we count nothing in 10 second. what is the count rate for 95% Confident Interval?

plug in the equation and give \lambda = 0.005 .

Now, we can see how the Confident Interval means. 95% means, in 100 measurement of 10 second interval, there is 95% that no count. thus, in 1000 second time interval, there is at most 5 count. which is same to say  \lambda = 0.005 .

or, if we put the  \lambda = 0.005 into the Poisson distribution.

P(0|0.005) = Exp(-0.05) = 0.951

this is same that it has 95% give 0 count.

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