# STD of Poisson distributed particle intensity

1. Nov 14, 2015

### theorem

1. The problem statement, all variables and given/known data
Say I have a detector that detects alpha-particles of some decay process. Let N be the amount of particles detected and let N be Poisson distributed. If I now define the intensity to be the amount of detected particles per second I = N/t, what would the standard deviation of I be? Also derive the standard deviation of ln I using propagation of error (error in t can be disregarded)..

2. Relevant equations
Poisson distribution

3. The attempt at a solution
If I let the amount of particles detected in a time interval t be n = μ/t, the probability of detecting x particles is given by the Poisson probability function with mean (parameter) μ = nt. The expectation value is therefore just μ and the std is sqrt(μ) = sqrt(nt). This means that the std for the intensity is sqrt(nt)/t = sqrt(n/t). That's the first part done.

As for the std of ln I, I don't really know what is meant by deriving it using propagation of error or maybe I'm just too tired. Any help is appreciated.

2. Nov 18, 2015

### Ygggdrasil

Shouldn't this be n = µt?

https://en.wikipedia.org/wiki/Propagation_of_uncertainty#Non-linear_combinations

Instead of defining a new variable µ in part 1, you should try applying the formula above to I = N/t in order to calculate sI in terms of N and t.