I need a python script$$

Exercise $2$: Maximum Likelihood and least squares

In the file exponential$\_$data.txt there are $1000$ decay time measurements between $0$ and $5$ s of muons, which have a mean lifetime of $=2s.$ The PDF used to generate this data is:

$P(t)=\frac{1}{*(1-e^{-\frac{5}{}})}*e^{-\frac{t}{}}$

where $$ is the mean decay time in $$ s and $t$ is the measured decay time.

$($a$)$ Calculate twice the negative$-$log$-$likelihood $(2*NLL())$ as a function of the lifetime $.$ Shift the values of the $2*NLL()$ such that the minimum is zero and plot it for $1\mathrm{.}8<<2\mathrm{.}2$ s$.$

$($b$)$ Bin the time measurements into a histogram of $40$ equally sized bins, from $0$ to $5s.$ Use the binned maximum likelihood method to calculate twice the negative$-$log$-$likelihood $(2*NLL())$ for these $40$ bins as a function of $.$ Shift the values of the $2*NLL$ such that the minimum is zero and overlay the plot with the unbinned calculation in part a$).$ Does it matter if you approximate the integral $($by using the mean of the bin$)$ of the function when computing the prediction for each bin $(f_i)?$ Argue quantitatively and$/$or calculate and compare.

$($c$)$ Now create a ${}^2$ function and calculate the ${}^2$ as a function of $.$ Plot it and overlay it on the NLL plot with the other two calculations in parts a$)$ and b$).$ Shift the ${}^2$ also such that the minimum is at zero. Compare the results.

$($d$)$ Bin the dataset into a wider histogram of $2$ bins and calculate the binned $2.$NLL and the ${}^2$ function again. How is the agreement now between them? How do they agree agree with the unbinned calculation in part a$)?$

$\backslash$table$[[$exponential$\_$data.txt$],[2\mathrm{.}36157335906],[4\mathrm{.}3340095117],[1\mathrm{.}54459209088],[1\mathrm{.}02583149632],[1\mathrm{.}52965487436],[3\mathrm{.}52931068162],[1\mathrm{.}09519676302],[0\mathrm{.}144951827224],[4\mathrm{.}55516567972],[0\mathrm{.}41844374904],[0\mathrm{.}844736157205],[1\mathrm{.}74170326767],[1\mathrm{.}53422145666],[1\mathrm{.}74237962417],[2\mathrm{.}8883874615],[3\mathrm{.}98074912863],[0\mathrm{.}536023747018],[1\mathrm{.}49082363197],[2\mathrm{.}81300952196],[1\mathrm{.}98147935073],[3\mathrm{.}35694304445],[0\mathrm{.}675999872838],[0\mathrm{.}359700293147],[4\mathrm{.}00953762641],[0\mathrm{.}700938858181],[0\mathrm{.}842091012328],[0\mathrm{.}0738584732608],[1\mathrm{.}11884965612],[1\mathrm{.}24849322578],[2\mathrm{.}61514639864],[2\mathrm{.}88810895792],[0\mathrm{.}154439759353],[3\mathrm{.}564019897],[0\mathrm{.}741363677707]]$