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Exercise $3$: Fitting with more than one parameter $($polynomial$)$

In the file polynomial$\_$data.txt you can find $30000$ measurements of the variable $x$ between the range $-1$ and $1.$

$($a$)$ Bin these measurements as a histogram with $20$ bins and calculate the uncertainty on the number of measurements in each bin using the Poisson distribution. Plot this histogram with error bars.

$($b$)$ Use the least$-$squares method scipy.optimize.curve$\_$fit to fit a polynomial of first, second, third and fourth degree to the histogram bins, taking the $x-$value to be the centre of each histogram bin and the $y-$value to be the number of measurements. Plot the resulting polynomial fits overlaid on the histogram.

$($c$)$ For each fit, calculate the uncertainties on the parameters using the covariance matrix returned by the scipy.optimize.curve$\_$fit method.

$($d$)$ Use the optimal fitting parameters to calculate the $\frac{{}^2}{n}df$ for the different fits and plot it as a function of the degrees of the polynomial.

$($e$)$ What do you think was the degree of the polynomial used to generate the data?

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polynomial$\_$data.txt