The code below will generate some locatons, x$1,$ data points, data$1,$ and error on the data, sigma$1.$

Using this data, answer the following questions:

a$)$ If you ignore the error and do a linear fit, what value of m and b do you get? What are the errors on each of these values?

b$)$ If you include the errors, what values of m and b do you get? What are the errors on these values? How close are these values to the ones used to create the data $($m $=2,$ b $=4)?$ Are the fitted values within $1-$sigma of the real values?

c$)$ Notice that the formula used to make the data also has an x$^2$ term. If you fit the data with a degree$-2$ polynomial, what values of the $3$ parameters do you get, and what is the error on each parameter? Are these values consistent with the values used to make the data?

d$)$ By using a degree$-2$ polynomial in part c$,$ how much better is the fit than in part b$)?$ You can determine this by comparing the calues of $.$ Is using the additional parameter reasonable when you compare the reduced $?$ Below is code to generate some data we saw in class, for a sin function plus some noise. In class we saw a decent fit to this, using:

p$=[1.,18.,0.]$ # initial guess bounds $=[[0,0,0],[$np$.$inf,np$.$inf,$2*$np$.$pi$]]$ # bounds on parameters popt,pcov $=$ optimize.curve$\_$fit$($sin$\_$fit,x$,$y$,$p$,$sigma$=$sigma,bounds$=$bounds,absolute$\_$sigma$=$True,method$=$'dogbox'$)$

However, we can also make the fitting fail.

a$)$ Try changing the initial guess to something else, say

p$=[1.,8.,0.]$

and rerun the fitting. What happens? Does this look like a good fit? Try changing your initial guess a few times to see roughly how close your guess has to be to get a good fit. Why does the fitting fail?

b$)$ How about if you try different methods for finding the best fit, i$.$e$.\text{'}$lm$\text{'},\text{'}$trf$\text{'},$ 'dogbox'? $($For $\text{'}$lm$\text{'},$ you'll need to take out the 'bounds' option.$)$ Do some methods find a good fit over a wide range than others?

c$)$ The optimize.minimize function has a wide range of methods available. First, write a new function that can be fed to optimize.minimize to find the best fit for this data. It should return the weighted for the parameters passed to it$.$

d$)$ Now use optimize.minimize on the function you wrote in part c$).$ Try out a few different methods with some 'bad' guesses. Do any methods produce a good fit even with a bad initial guess? You have some substance you suspect is radioactive. With a Geiger counter, you record the following counts in $1$ second intervals spaced $1$ minute apart:

time $($minutes$)$: $[0.1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23.24.25.26.27.28.29.30.]$ counts $($per second$)$: $[131901005573555858473734364332302825172520203023272211219101821]$

a$)$ Fit a reasonable function $($An expoential decay$)$ to this data, using curve$\_$fit. Assume the error $($sigma$)$ on each measurement is $1$ count$)$ What is the decay constant of your substance? What are the errors on this parameter?

b$)$ You should also be able to fit this data by making a change of coordinates, and then fitting a line to the new data. Do this. What decay constant do you get, and what are the errors? Is this consistent with your fit from part a$)?$

c$)$ Really, the count data will have poisson errors. Assuming the counts were measured in a single $1-$second time interval, the error $($sigma$)$ will be the square root of the number of counts. Using this, try using curve$\_$fit again and see what you get for the decay constant ant error on this parameter.

d$)$ To make you fit better, keep in mind there are probably some background counts as well $($from cosmic rays, electronic noise in your instrument, etc.$).$ Try fitting again using and exponential decay plus a constant $(3$ parameters$)$ with and without assuming Poisson noise. How do these results compare to the values in part a$)$ and c$)?$