PYTHON

Suppose you needed to generate a Gaussian $($normal$)$ distribution

$(,|)=\frac{1}{\sqrt[\phantom{2}]{2}}{e}^{-\frac{(x-{)}^2}{2}{}^2}$

of $N_{RV}$ random variates, but the only random number generator you know about

is numpy.random.uniform $(0,1,N).$ However, assume you do know about the central

limit theorem. Create a Python script

gengauss.py which does the following:

Reads $,$ and $N_{RV}$ from the command line. Be sure that your script checks that

it has the proper inputs, and prompts the user if not.

Uses the central limit theorem to generate $N_{RV}$ values of $x$ having the proper

distribution, using only random numbers from numpy. random. uniform $(0,1,N).$

Creates a plot in the file gengauss.png$,$ showing a histogram of $x$ values with

the PDF $(,|)$ superimposed.

Be sure the distribution you generate has mean and width as specified on the command

line.

The distribution of decay times for elementary particles is well$-$modeled by the expo$-$

nential distribution, with PDF given by:

$(|)=\frac{1}{}{e}^{-\frac{t}{}}$

$t$ is the decay time and $$ is the lifetime parameter of the elementary particle.

The file muons $20051130.$ dat contains data from an experiment to measure the life$-$

time of the elementary particle known as the muon. $($A similar experiment lives in

South Physics, and if you take PHYS $3719$ you may have an opportunity to perform

this experiment.$)$ The file contains a single column with the decay time, in microsec$-$

onds, of a set of cosmic ray muons recorded on November $30,2005.$

For this problem, you are to determine the muon lifetime, with uncertainties and

goodness$-$of$-$fit using the ${}^2$ minimization method. You should create a Python script

muon$-$chi$2.$py to perform the following:

Open the data file and sort decay times into bins. I recommend using $25$ bins

between $0$ and $8sec.$

Create independent variable $x$ from the central value of the bins, and dependent

variable $y$ from the number of entries in each bin. Use the square root of the

number of entries as the $($assumed Gaussian$)$ random error in the dependent

variable.

Use scipy.optimize.curve$\_$fit to extract the best fit value for $.$ Compute

$\frac{{}^2}{d}f$ and goodness of fit. Print your results to standard output.

Submit your Python script

muon$-$chi$2.$py$.$