The central limit theorem roughly states that a large collection of identical random variables behaves like the normal distribution. Many investigations into the eigenvalues of random matrices suggest experimentally that this statement holds, i. e, the eigenvalues of matrices whose elements are not normal behave, more or less, like the eigenvalues of normally distributed matrices It is of value to note that the normal distribution with mean u and variance o has Pr(t) 2 Univariate hi Isograms m数2 In Figure 2, we plot the plot the normal distribution Figure 1: This figure illustrates the idea that the probability density is a histo Code 1 is our MATLAB code to obtain this figure >>a= randn(1,5000);[n,x]=hist(a,[-3:.2:3]); >>bar(x,n/(5000*.2)) hold on, plot(x, exp(-x. 2/2)/sqrt(2*pi)), hold offThe central limit theorem roughly states that a large collection of identical random variables behaves like the normal distribution. Many investigations into the eigenvalues of random matrices suggest experimentally that this statement holds, i.e., the eigenvalues of matrices whose elements are not normal behave, more or less, like the eigenvalues of normally distributed matrices. It is of value to note that the normal distribution with mean µ and variance σ2 has 1 px(t) = σ √2π e−(x−µ)2/2σ2 . 2 Univariate Histograms In Figure 2, we plot the normal distribution as well as a histogram obtained from 5000 samples from the normal distribution We see in the second line of the code below that we divide the counts n by the total number times the bin size: 5000*0.2. This guarantees that the total area of the boxes over the whole line is normalized to 1. -4 -3 -2 -1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Figure 1: This figure illustrates the idea that the probability density is a histogram Code 1 is our MATLAB code to obtain this figure. >> a=randn(1,5000);[n,x]=hist(a,[-3:.2:3]); >> bar(x,n/(5000*.2)); >> hold on,plot(x,exp(-x.^2/2)/sqrt(2*pi)),hold off