# Simulating photons on detector statistically

I am simulating a simple detector, imaging a very faint astronomical source (so that some dozens of (uncorrelated) photons are detected in every frame)

Currently I am first sampling from Poisson-distribution to determine how many photons are to arrive in a given frame.

Then for every photon I have to determine the pixel it will hit: I have for every pixel a number, that gives the probability, that if a photon comes, it will hit that pixel. (so that the sum over all pixels is 1.) I have tools to sample N hits from this PSF (the pixelwise hit probabilities are proportional to the Point Spread Function of the telescope)

It would however be more convenient, if I could simply multiply the PSF with the expected number of photons per frame, and sample for each pixel from this Poisson distribution.

Are the two methods equivalent? If not, which of them is more correct?

• From what I read you have all the tools to decide this numerically. – my2cts Jan 7 at 17:11
• The coefficient of linear expansion $\alpha$ is defined as $\ell + \Delta\ell = \ell(1 + \alpha)\Delta T$ – Jack Rod Jan 7 at 17:23

If $$X$$ is a random variable, which is distributed according to a Poissonian distribution with rate parameter $$\lambda_1$$, we write $$X\sim \textrm{Pois}(\lambda_1)$$. Now, if $$Y\sim \textrm{Pois}(\lambda_2)$$ then $$Z=X+Y\sim \textrm{Pois}(\lambda_1) + \textrm{Pois}(\lambda_2) = \textrm{Pois}(\lambda_1 + \lambda_2)$$ In your case you have the same rate parameter $$\lambda$$ for each pixel, $$X_{\textrm{pixel}} = \textrm{Pois}(\lambda)$$. Thus, if you have $$N$$ pixels per frame the photon number per frame is distributed as $$X_{\textrm{frame}} = \textrm{Pois}(N\lambda)$$. Hence, you are definitely allowed to sample from a Poissonian distribution for each pixel, if you assume that the counts on each pixel are statistically independent.