Untitled Python Project

import numpy as np import matplotlib.pyplot as plt

def sample(n): # extra n random 0's or 1's, multiply by 2, then subtract by 1 to get -1 or 1 return 2*np.random.randint(2,size=n)-1

def sample_mean(n_obs,n): return [np.mean(sample(n)) for _ in range(n_obs)]

data = [sample_mean(1000, n) for n in [10,100,1000,10000]]

# plot histograms for 1b fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(nrows=2,ncols=2) fig.suptitle('Histograms for Various n') # give specifics for each subplot ax1.set_title('10 Samples') ax1.set_xlim([-1, 1]) ax1.set_xticks([]) ax1.hist(data[0], density=True) ax2.set_title('100 Samples') ax2.set_xlim([-1, 1]) ax2.set_xticks([]) ax2.hist(data[1], density=True) ax3.set_title('1000 Samples') ax3.set_xlabel('Sample Average') ax3.set_xlim([-1, 1]) ax3.hist(data[2], density=True) ax4.set_title('10000 Samples') ax4.set_xlabel('Sample Average') ax4.set_xlim([-1, 1]) ax4.hist(data[3], density=True) plt.show()

data_sqr = np.square(data) var = [np.mean(n) for n in data_sqr] logvar = np.log10(var) #generate plot x values (log10 of 10, 100, 1000, 10000) xes = [np.log10(x) for x in [10, 100, 1000, 10000]]

plt.plot(xes, logvar, 'o') plt.xlabel(r'$log_{10} (n)$') plt.ylabel(r'$log_{10} (var(\bar{x}))$') plt.show()

xline = np.arange(1, 4.05, 0.05) fit = -xline plt.plot(xes, logvar, 'o') plt.xlabel(r'$log_{10} (n)$') plt.ylabel(r'$log_{10} (var(\bar{x}))$') plt.plot(xline, fit) plt.show()

def probs(data, n_bins): # use numpy's histrogram function to separate data into finite pieces of x ns, bins = np.histogram(data,bins=n_bins) # this line returns the midpoint of each bin to serve as the x values for each plot below bins = bins[:-1] + 0.5 * (bins[1:]-bins[:-1]) # taking the number of sample means in each bin, dividing by the total number of samples # and multiplying by bin size, we get the probability density we need ps = ns/np.sum(ns)*(bins[1]-bins[0]) return bins, ps

xs = np.linspace(-1,1,100) plt.plot(xs,-xs**2/2) xs, ps = probs(data[0], 20) plt.xlabel(r"$\bar{x}_n$") plt.ylabel(r"$n log(p(\bar{x}_n))$") # for each value of n, take the logarithm, shift the maximum to 0 and divide by n plt.plot(xs, (np.log(ps)-np.max(np.log(ps)))/10, "o", label='n=10') xs, ps = probs(data[1], 20) plt.plot(xs, (np.log(ps)-np.max(np.log(ps)))/100, "o", label='n=100') xs, ps = probs(data[2], 20) plt.plot(xs, (np.log(ps)-np.max(np.log(ps)))/1000, "o", label='n=1000') xs, ps = probs(data[2], 10) plt.plot(xs, (np.log(ps)-np.max(np.log(ps)))/10000, "o", label='n=10000') plt.legend() plt.show()

def make_gaussian(x, mu, sigma_sqr): prefactor = 1/np.sqrt(2*np.pi*np.sqrt(sigma_sqr)) exponent = -(x - mu)**2 / (2*sigma_sqr) rho = prefactor * np.exp(exponent) return rho

x = np.arange(-1, 1.01, 0.005) curves = [make_gaussian(x, 0, i) for i in [0.1, 0.01, 0.001, 0.0001]] # initialize 4 subplots fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(nrows=2,ncols=2) fig.suptitle('Gaussians for 2a') ax1.set_title('n=10') ax1.plot(x, curves[0]) ax1.set_xticks([]) ax2.set_title('n=100') ax2.plot(x, curves[1]) ax2.set_xticks([]) ax3.set_title('n=1000') ax3.plot(x, curves[2]) ax4.set_title('n=10000') ax4.plot(x, curves[3]) plt.show()

m = np.linspace(0, 1, 100) #a different way to generate evenly spaced points on an integral m = m[1:-1] # this removes problem values of 0 and 1 where the logs will calculate infinities I = -((1-m)*np.log(1-m)+m*np.log(m)+np.log(2)) plt.plot(m, I) plt.xlabel(r'$m_+$') plt.ylabel(r'$I~(m_+)$') plt.show()