TOPIC = CENTRAL LIMIT THEOREM. CLASS = PROBABILITY ONLY NEED A MATLAB CODE. PLEA

Question

TOPIC = CENTRAL LIMIT THEOREM.

CLASS = PROBABILITY

ONLY NEED A MATLAB CODE. PLEASE IF YOU HAVE KNOWLEDGE ABOUT PROBABILITY AND RANDOM SIGNAL AND MATLAB HELP ME.

MANY THANKS FROM NOW ON.

1) Create N random variables (X XN) each with L samples. These should be independent and identically distributed. Generate Sn, which will be the sum of these random variables. 2) Find the mean and variance of X e. Hand o2. Since the RV's are iid's, H will be the mean of all the other RV's in this set too. You can verify this by finding the mean of a few other variables 3) Create the random variable Zn by adding N random variables as nH. 4) Plot the pdf of Zn 5) Repeat steps 3 and 4 for N-10, N-50 and N 100. For each case, repeat for values of L 100, 1000, 10,000 and 1000,000 6) Comment on the effects of the various values of Land N. 7) Repeat for each of the three RV's listed below: Uniform random variable Exponential random variable (lambda 0.5) Gaussian random variable (mu 5, sigma 2)

Explanation / Answer

clear;

N_die=6; % 6 sided die

N_roll=30; % 30 roll ensembles

range=[1,N_die];

disp('Input N')

N = input ('N: '); % total ensembles

subplot (2,2,1)

for m = 1:N;

% L1=randint (1,N_roll, range); % first way to get random numbers

L1=round (1+(N_die-1)*rand (N_roll, 1)); % second way for random numbers

% hist(y,nb) draws a histogram with nb bins.

hist(L1)

axis([0 7 0 12]) % sets scaling for the x- and y

title ('Random Distribution of 6 sided die numbers')

xlabel('Random Integer', 'FontSize',8)

ylabel('Frequency', 'FontSize',8)

%num2str (m,p) converts the value of m to characters with p digits, cat(2,'a','b') concatenates a and b

L5 (m)=mean (L1); % contains the mean of the L1 distribution

str=cat(2, 'Mean (',num2str(m,4),') = ',num2str(L5(m),4));

%place the the value of the time and its mean on the screen, in red

text(2,11,str,'Color','b');

%Change the color of the graph so that the bins are red and the edges of the bins are white.

h = findobj(gca,'Type','Patch'); % change bin colors

set(h,'FaceColor','r','EdgeColor','w')

pause(0.5);

end

%L5; % at this point L5 is a distribution of means

%figure

%pause(2)

subplot(2,2,2)

hist(L5); % histogram

title('Distribution of Means, and a Gaussian')

xlabel('Mean','FontSize',8)

ylabel('Frequency','FontSize',8)

h = findobj(gca,'Type','Patch'); % change bin colors

set(h,'FaceColor','b','EdgeColor','w')

hold on

ML=mean(L5);% this is the mean of the means

ST=std(L5); % standard deviation of the means

a=min(L5)-2.0;

b=max(L5)+2.0;

dx = (b-a)/100.0;

x = [a:dx:b];

y =max(hist(L5))*exp(-(x-ML).^2);

plot (x,y,'m:')

% We next find the z-scores of each mean and plot these versus the means

% We should get a straight line for a normal distribution => positive correlation

pause(1)

%figure

subplot(2,2,3)

z=(L5-ML)/ST; % z-scores

% Also let's superimpose a line on this graph

a=min(L5)-1;

b=max(L5)+1;

dx=(b-a)/100.0;

x = [a:dx:b];

par=polyfit(L5,z,1); % actual line fit parameters

str=cat(2,'fit:slope=',num2str(par(1))); %num2str(t) converts the value of par91) to characters, cat(2,'a','b') concatenates a and b

y=polyval(par,x);% actual line fit

plot (L5,z,'bo',x,y,'r:') % the z-scores plotted versus the means

legend('data',str,4);

grid on

title('Correlation of the means and their Z-scores')

xlabel('Means','FontSize',8)

ylabel('Z-Scores','FontSize',8)

h = findobj(gca,'Type','Patch'); % change bin colors

set(h, 'FaceColor','b','EdgeColor','w')

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