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Cell["N17: \[Mu] lifetime", "Title",
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The purpose of this lab is to measure the muon lifetime by plotting the \
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plot it using the Plot program, I don't think it's really necessary for ",
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We now use the FindFit function to fit to a nonlinear function. The Fit \
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This isn't bad for an estimate. We expect something like 20 ( = 2\[Mu]s), so \
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We would like to do a ",
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We shall take the ratio of positive to negative muons to be a constant.\
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We can estimate the background using the singles counts. We assume the \
singles counts is much greater than the subset of muon candidates (doubles). \
We can read off the total number of singles from summing our counts and we \
can read off the total time in the dataout file (it's the last number of the \
file). Note that the dataout files also have a \"total count\" before this \
number, but htis seems to be off by roughly a factor of two. 

The singles count has a large error, maybe on the order of (plus or) minus \
1000, but this is fine for estimating the rate.\
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Cell["\<\
I'll estimate the background using the Poisson distribution. The probability \
for a double event, given a single event, is\
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I feel like this is a really rough estimate, but it should be good enough as \
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Cell["\<\
So the naive expectation is that each bin will have 2.9 background events. \
This looks a bit high given my data, but I guess I'll work with this for a \
while. Anyway, the maximum likelihood (ML) analysis should catch the right \
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The below is just a parenthetical note about how to get the total number of \
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Construction of Likelihood and Log-Likelihood function: 1 parameter\
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 "Let's start with a simple example of a one parameter fit. This is just ot \
get a feel for using maximum likelihood. We assume: no background, one fit, \
effectively infinite detector. Steps...\n1. ",
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 ". Define the distribution function and evaluate the product of N such \
functions according to the observed data. It might be easier to start with \
the log of the likelihood function since this is relatively straightforward \
to do analytically.\n2. ",
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 ". This may be nontrivial, but I can limit my search to points near where I \
expect to find the maximum. I should be able to use ",
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 "'s built in optimize function, or otherwize I can just do a stupid scan.\n\
3. ",
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R is the ratio N+/N- of positive muons to negative muons at sea level. This \
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My new strategy will be to calculate the log likelihood funciton. Since the \
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could shift the maxima significantly.

Anyway, if I end up using this approximation for my final analysis, I \
probably ought to include it in the theoretical error. This would require a \
bit of fancy footwork with calculations, but it should be do-able.\
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I've used the Stirling approximation for the Log Likelihood Function.\n2. \
I've APPROXIMATED the maximum of this \"Stirling Log Likelihood\" by plugging \
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values of the parameters are stable. \n\nConcerns:\n1. I should be VERY \
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approximation. (Since Stirling's holds well for large N and since there's \
lots of data even in the large time bins, this shouldn't change \
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So apparently I was doing something wrong above since I should be doing a \
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Cell["\<\
Well, that was suprisingly quick. Anyway, below are just some examples of \
what I can do with this.\
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Cell["\<\
I've recorded all the useful information into new variable names. By the way, \
here's the variation for \
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Cell["\<\
For the errors we can make the assumption that N is sufficiently large that \
we can trat the distribution about the mean as Gaussian. Then each standard \
deviation is given by the contour that gives (max-stddev/2) log likelihood. \
In particular, the 1 std dev error is the contour that gives the maximum log \
likelihood minus a half.\
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First, let's try to estimate the 'theory error' from the value of R.\
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