# How to calculate poisson uncertainties when the observation windows change duration

1. Apr 3, 2012

### kop442000

I am trying to plot the flux from an Astrophysical source as a function of time. Due to the nature of the source, I am only receiving a handful of photons in each time bin.

So imagine I had 10 observing periods of 10 days each, in which my telescope received the following number of photons:

10,9,7,12,5,7,8,7,5,10

Now if I wanted to get the uncertainties on these measurements, I know that I could use poisson statistics and have:

10±√10, 9±√9, 7±√7 etc.

So I think I am okay up to here. But I need to modify my numbers, because in the first 10 day observation, the instrument was looking at the source for slightly longer than the next 10 day period. In fact, the period that the telescope was looking at the source was slightly different in each 10 day bin.

So to correct for this, I simple divide the number of photons by the observing time, which gives me an average flux over the 10 day period:

10 photons divided by 93 minutes (say) cumulative observation time over the 10 days would give a flux of 0.11 photons / minute
9 divided by 71 minutes gives a flux of 0.13 photons / minute

and so on. So the flux for the second bin is bigger because the source was observed for a significantly shorter time period.

So I am eventually getting to my question:

Because of this added complication of different observing lengths, I'm now not entirely certain how I get my uncertainties. I am assuming I would use the poisson statistics again, but how do I modify them to fit in with my units and take into account the different observation lengths.

For example if I had two 10 day periods that had each registered 10 photons, but in one, the source was monitored for twice as long as the other, one would have twice the flux that the other has, but if I ignored the observation times, their uncertainly would just both be √10 (divided by their respective observation lengths). Is it just as simple as this??

Hope this is at least somewhat clear!

Thank you in advance of any help :-)

2. Apr 4, 2012

### Stephen Tashi

Re: How to calculate poisson uncertainties when the observation windows change durati

I think the lack of reponses to your post is due to an uncertainity about what you mean by "uncertainties". I've noticed that people in the physical sciences often use "uncertainty" to mean standard deviation. Another meaning of "uncertainty" has to do with the significant figures in a measurement. Another meaning of "uncertainty" is Shannon entropy. Some people use uncertainty to mean something like "confidence interval" and often mistakenly think that a "confidence interval" has the properties of a "credible interval".

I'll assume, for the time being that your "uncertainty" means "standard deviation". A "standard deviation" can be (at least) 3 different things. 1) The standard deviation of a random variable 2) The standard deviation of a sample 3) The estimator of the standard deviation of a distribution computed from a sample.

Each of those three things can be talked about in two ways. We can talk about them as random variables themselves - for example, we can talk about the distribution of the sample standard deviation. Or we can talk about them as a particular realization of a random variable - for example, we might say that the sample standard deviation is 38.2.