Create N random variables X1 XN each with L samples These s

Create N random variables (X_1, ..., X_N) each with L samples. These should be independent and identically distributed. Generate S_n, which will be the sum of these random variables. Find the mean and variance of X_1. i.e. mu and sigma^2. Since the RV\'s are iid\'s, mu 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. Create the random variable Z_n by adding N random variables as: Z_n = S_n - n mu/sigma squareroot n Plot the pdf of Z_n 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. Comment on the effects of the various values of L and N. 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) Type a report on your findings. Make it professional!

Solution

X1=rand(1,L)

Since there are N random variable, the random numbers can be generated by

X=rand(N,L);

The sum of random variable:

sn=sum(X,2);   

2 ) The mean of X1 can found by

mu=mean(X(1,:));

disp(mu);

the mean of X1,X2,X3.. Xn can be found by

mu=mean(X,2);

the variance of X1,X2, X3.. Xn

sigma=var(X,0,2);

// MATLAB CODE

L=10; % no of samples in Random variable

N=10; % no of random variable

X= rand(N,L);   % random numbers

sn=sum(X,2);    % sum of random variable

mu=mean(X,2);   % mean of X

sigma=var(X,0,2); % finding variance

zn= (sn-(mu.*9))/(sqrt(sigma).*sqrt(N)); % finding Zn

plot(zn);