How Is Muon Lifetime Estimated Using Maximum Likelihood Estimation?

[Yoshimoto]

Homework Statement

As a lab experiment, we must try to measure and calculate the lifetime (τ) of a muon. After all data is collected, the lifetime can be calculate by using the maximum likelihood estimation.
The function on which MLE is applied:
http://j.imagehost.org/view/0139/Schermafbeelding_2010-04-27_om_22_29_38

One of the goals of this experiments is to do the MLE manually. For that I solved al the equations that are given by the definition of MLE. I finally ended with this equation:
http://j.imagehost.org/view/0926/Schermafbeelding_2010-04-27_om_22_29_56
T_a, T_b and N are constants, T_i is the dataset and Tau is the lifetime (this should be calculated).
I know this equation is correct. It should now be solved with mathematical software (I prefer matlab).

Homework Equations

My problem is to solve this in matlab. When i try to do this, it gives me the error:
"The expression to the left of the equals sign is not a valid target for an assignment."

The Attempt at a Solution

This is the code i used:

n=61565;
ta=402e-9;
tb=12e-6;
tau=solve(-n/x*(-n(tb*exp(-tb/x)+ta*exp(-ta/x))/(exp(-tb/x)-exp(-ta/x)))+sum(ti/x)=0,x);

T_i is a variable with 61565 rows and one column

Does anyone know what I am doing wrong?

Thanks!

 

[mighty2000]

Dear fellow scientist,

Thank you for sharing your progress on the MLE calculation for the lifetime of a muon experiment. It seems like you have a good understanding of the equation and have successfully derived it. However, the issue you are facing with solving it in matlab may be due to the syntax of the solve function.

The solve function in matlab requires the equation to be in the form of "expression = 0". In your code, you have already set the equation equal to 0, but you also have an additional term (sum(ti/x)) on the left side of the equation. To fix this, you can subtract this term from both sides of the equation, so that it becomes:

-n/x*(-n(tb*exp(-tb/x)+ta*exp(-ta/x))/(exp(-tb/x)-exp(-ta/x))) = -sum(ti/x)

Then, you can use the solve function as follows:

tau = solve(-n/x*(-n(tb*exp(-tb/x)+ta*exp(-ta/x))/(exp(-tb/x)-exp(-ta/x))) + sum(ti/x) - (-sum(ti/x)), x);

This should give you the value of tau as a symbolic expression. If you want to convert it to a numerical value, you can use the double function:

tau = double(tau);

I hope this helps and good luck with your experiment!