# Poisson distribution and muon decays

#### cohara

##### New Member
Hello,

I have gathered muon decay data and used R's non linear least squares function to find the mean muon lifetime. I did this by taking the exponential part of the decay curve and solving for the inverse of the lifetime, where the function is Nt=N0(exp(-t/τ)+c. That bit is fine.

The thing is that I have read in numerous source that muon decay follows a poisson distribution (and it looks like it does), but I cannot figure out what the parameters are.

The mean muon lifetime that I have determined is 1.74µs. I know that that is not correct but that is ok given my dataset. In R and Matlab I have tried to model the decay curve that I am getting but to no avail. For a start I cannot enter anything other than integer values for my quantiles. I know that this is because the poisson function is designed for count data, so a non-integer value makes no sense, but then i don't understand how a poisson function can describe muon decay, since a non-integer muon decay time makes perfect sense. This leads me to think that maybe the mean decay time is not the mean value that I should be using (lambda in the dpois function in R). I have tried writing my own function in R to carry out the calculation, but of course the factorial cannot be used with anything other than integers, which, again, makes sense.

I have used integer values of between 1 and 20 for my quantiles with a mean of 1.74 (simulating microseconds) but the distribution does not match my experimental one at all. I have tried 1/1.74 as my mean value, because it at least gives a value per second (somewhat akin to a rate - although it is not a rate) and again used 1 to 20 as my quantiles. This actually seems to fit, but I don't think that it is correct; it just doesn't seem right to me because the mean decay time should surely describe the function more accurately, and when I use 1/1.74 as the mean I get a 67% chance of a decay occurring at time 0, with it dropping from there, while my experimental data shows that the probability at time = 1µs is greater.

I have read around this as much as I can but no source has helped me to understand the problem. It seems to me that the mean (lambda in R) should be 1.74µs with microsecond quantiles, and that then the distribution should resemble the experimental one that I got, but I'm not sure.

Any help understanding this would be greatly appreciated.

#### BGM

##### TS Contributor
I have not read your post in detail but just my two cents:

In a Poisson process, the inter-arrival time, i.e. the time between two consecutive events follows an exponential distribution. And therefore the support is just need to be positive real numbers.

If you fix a time interval, then the number of arrivals / events occur within the interval will follows a Poisson distribution, and the support will be non-negative integer.