Problem 3 (3 points) Random Walk theory provides us with the following nice (anf maybe counterintuitive) asymptotic statements C2N lim P lim P ANS arcsin(/x) lim P | x| =-arcsin(y) We say that the random variables C2w/2N, L2N/2N, M2N/2N converge in distribution to the Arcsine Distribution The interesting property about the Arcsine distribution is that its density (see its formula above) is U-shaped on (0,1). In other words, if X is arcsine- distributed on (0, 1), the probabilty that X takes very small values near 0 or very large values near 1 is rather high, but the probability for taking values around, say, 0.5, is low xnp. linspace (arcsine.ppf (0.05), arcsine.ppf (0.95), 100) plt.title("Density of the arcsine distribution") plt.plot(x, arcsine.pdf(x), linewidth-2, color-'b' 3.5 3.0 2.5 2.0 15 For 2N 1000 sample 10,000 realisations of each of the random variables C2N/2N, L2N/2N, and M2N/2N, respectively. Display a normalized histogram for all three simulations, along with the probability density function of the arcsine distribution, to check the above facts numerically! [ ]: ### Implement your simulations here : ### Complete the plot commands accordingly c-arcsine.rvs(size-10000) # This has to be replaced by the simulated values for C-2N plt.figure(figsize (10,5)) plt.title("Normalized histogram for 10000 realisations of $C_t1000 $") plt.hist(c, bins-'auto', normed 'True) plt.plot(x, arcsine.pdf(x), linewidth-2, color-'r', label-"true arcsine density") plt.legend) 1 = arc sine.rvs (size: 10000) # This has to be replaced by the corresponding simulated values for L_TN plt.figure(figsize (10,5)) plt.title( "Normalized histogram for 10000 realisations of $L 11000 $") plt.hstCL, bins-'auto', normed='True') plt.plot(x,arcsine.pdf(x), linewidth-2 , colors' r'' label=" true arcs ine density") plt. legend) m-arcsine.rvs(size-10000) # This has to be replaced by the corresponding simulated values for M-2N plt.figure(figsize-(10,5)) plt.title("Normalized histogram for 10000 realisations of $M_11000)s") plt.hist(m, bins 'auto,normed 'True') plt.plot(x, arcsine.pdf(x), Linewidth-2, color 'r', label-"true arcsine density") plt. Legend) matplotlib. legend.Legend at 0x118916390 Problem 3 (3 points) Random Walk theory provides us with the following nice (anf maybe counterintuitive) asymptotic statements C2N lim P lim P ANS arcsin(/x) lim P | x| =-arcsin(y) We say that the random variables C2w/2N, L2N/2N, M2N/2N converge in distribution to the Arcsine Distribution The interesting property about the Arcsine distribution is that its density (see its formula above) is U-shaped on (0,1). In other words, if X is arcsine- distributed on (0, 1), the probabilty that X takes very small values near 0 or very large values near 1 is rather high, but the probability for taking values around, say, 0.5, is low xnp. linspace (arcsine.ppf (0.05), arcsine.ppf (0.95), 100) plt.title("Density of the arcsine distribution") plt.plot(x, arcsine.pdf(x), linewidth-2, color-'b' 3.5 3.0 2.5 2.0 15 For 2N 1000 sample 10,000 realisations of each of the random variables C2N/2N, L2N/2N, and M2N/2N, respectively. Display a normalized histogram for all three simulations, along with the probability density function of the arcsine distribution, to check the above facts numerically! [ ]: ### Implement your simulations here : ### Complete the plot commands accordingly c-arcsine.rvs(size-10000) # This has to be replaced by the simulated values for C-2N plt.figure(figsize (10,5)) plt.title("Normalized histogram for 10000 realisations of $C_t1000 $") plt.hist(c, bins-'auto', normed 'True) plt.plot(x, arcsine.pdf(x), linewidth-2, color-'r', label-"true arcsine density") plt.legend) 1 = arc sine.rvs (size: 10000) # This has to be replaced by the corresponding simulated values for L_TN plt.figure(figsize (10,5)) plt.title( "Normalized histogram for 10000 realisations of $L 11000 $") plt.hstCL, bins-'auto', normed='True') plt.plot(x,arcsine.pdf(x), linewidth-2 , colors' r'' label=" true arcs ine density") plt. legend) m-arcsine.rvs(size-10000) # This has to be replaced by the corresponding simulated values for M-2N plt.figure(figsize-(10,5)) plt.title("Normalized histogram for 10000 realisations of $M_11000)s") plt.hist(m, bins 'auto,normed 'True') plt.plot(x, arcsine.pdf(x), Linewidth-2, color 'r', label-"true arcsine density") plt. Legend) matplotlib. legend.Legend at 0x118916390
