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PHYCS 3730/6720 Lab Exercise

Reference
Answer file **Mylab20.txt**.

#### Exercise 1 Fitting your radioactive decay curve.

In assignment 6 you created a Monte Carlo model of radioactive decay
with your code **decay.py**. In this exercise, you are asked to
use the **scipy.optimize** **curve_fit** function to fit your
results from exercise 2 for decays vs time to determine the best fit
lifetime and initial number of atoms. (You know what input values
were used when you ran the model, but the exercise here is to pretend
you don't know them, and see how well you could have estimated them by
fitting the curve.)
To use **curve_fit** you need a list of times, number of decays at
those times, and a standard deviation for the number of decays. You
have the first two from the exercise. We know the probability
distribution is Poisson, and when the number of counts is high enough,
the Poisson distribution looks somewhat Gaussian and has a standard
deviation approximately equal to the square root of the number of
counts. On the other hand, when the number of counts is small, as we
have seen, the Poisson is not symmetric like a Gaussian. For a better
approximation, we add 1 to the number of decays at every point. Also,
we use the square root of the new number to estimate the standard
deviation of that point. Finally, let's avoid having too many decay
values close to zero, where this approximation is worst. so fit only
the first 60 values in your data.

Write a short Python script that reads the decay results from
assignment 6, exercise 2, computes the standard deviations, and then
fits the function, See the Scipy curve fitting handout linked above
for a similar example. Note, however, that here you are not creating
artificial data. You are using your results from Assignment 6. And
your function is for the number of decays versus time, not the number
of surviving atoms versus time, as in the handout.

def f(t,N,tau):
return dt*N/tau*np.exp(-t/tau)

for the best values of **N** and **tau**. (Here **dt = 5**.)
Choose initial values for the fit parameters that are not too far from
the values that you know.
In the answer file, paste your Python script and the result from your
fit, identifying the best fit values.

#### Exercise 2. Interpreting the covariance matrix

Use the covariance matrix to determine the standard deviations in the
best fit values of **N** and **tau**. How close did you get to
the true values? Answer by saying by how many standard deviations you
miss them for each value. Then determine the correlation coefficient
between **N** and **tau**. What does a positive correlation
mean?