Statistical error analysis of Geiger counter?

[carnivalcougar]

1. Homework Statement

An average value of N measurements is defined as:

ravg = 1N ∑ ri where i = 1 and it sums up to N measurements

By using this expression in the master formula for a general function z = (x1, x2,...) error propagation, we find the δravg error propagation:
(δravg)2 = ( (∂ravg (ra, r2,...) / ∂r1 ) (δr1))2 + ( (∂ravg (ra, r2,...) / ∂r1) (δr1))2 + ...

Finally, we arrive at the following conclusion:
δravg = δrN√
since all measurements are independent (δr1)2 = (δr2)2 = ... = (δr)2. In your present investigation the counts of the geiger counter are integer numbers. They can become zero, navg = 0, if the shiled is thick or the time interval becomes small. Does this mean that standard deviation vanishes, \sqrt{navg} --> 0, and statistical error becomes infinite, δnavg ∝ navg-1/2 → ∞ ? Or, does the presented statistical error analysis become inapplicable in this case? Is it better to have one longer time interval or many short intervals of the same cumulative duration to minimize the standard deviation (or does it make no difference)?



2. Homework Equations
Provided in question


3. The Attempt at a Solution

I think that the error analysis becomes inapplicable. I don't think the standard deviation vanishes and the statistical error becomes infinite. This may be completely wrong though. I'm not sure if the interval matters though.

 

[BvU]

I see two relevant equations, one for r_avg (where I assume you mean ravg = 1/N ∑ r_i) and one for δr_avg. Ok.

I don't see where δravg = δrN√ comes from . Can you complete it and work it out ?

I also don't see why all δr_i should be equal. What do you measure ? Do you use equal time intervals for the measurements ?

Did you learn about Poisson statistics already ?

 

[dauto]

I think if you want people to take the time to help you, you ought to at least take the time to make sure your equations are written correctly.

 

[carnivalcougar]

I attached pictures of the question with the equations typed out so they will definitely be correct.

We also did not learn Poisson processes.

 

[BvU]

I see. So what I assumed was your attempt at a solution was really the original problem statement.
No relevant equations at all ? No attempt at a solution, except some thoughts ?

And apparently, you/they use equal time intervals for the measurements, so the errors might be the same for each r_i.

So where are you with your present investigation ?

What they are also asking is "If you have an hour of counting time, is it better to take 60 measurements of 1 minute, 30 of 2 or 12 of 5 or 1 of 60 minutes, or doesn't it matter ?"

What is observed with a Geiger counter (the name says it already) is counts. Rate = counts/time interval, and those we had fixed. So with a given rate r and a given N you can say something about the expected number of counts per interval.

If you are not familiar with Poisson, then: do you know about binomial distributions ? Gauss distribution ?

 

[Curious3141]

We also did not learn Poisson processes.

You will, of course, recall that I asked the same question that BvU is asking when you posed another question on radioactivity in a different thread in Intro Physics.

You have to learn about Poisson processes (and the approximation of the Poisson distribution to the normal distribution for large mean values). Whether you've been formally taught these things or not seems to be irrelevant if they're asking you questions on these topics.

I suggest self-study - the net is a great resource.