Consider and random variable RV X that is exponentially dist

Consider and random variable (RV) X that is exponentially distributed with

E[X]=5. Using MATLAB, do the following:

a. Generate 1,000,000 samples of this random variable.

b. Plot in figure 1 the histogram of the generated samples (bar chart) and

include the labels.1

c. Plot in figure 2 the p.d.f of the generated samples and compare it to the

theoretical p.d.f of exponential R.V. In your plot use the following

settings:

i. Use black line for the theoretical p.d.f.

ii. Use red diamonds for the experimental one.

iii. Include axes labels, title and legend.

d. Plot in figure 3 the c.d.f of the generated samples and compare it to the

theoretical c.d.f of exponential R.V. In your plot use the following

settings:

i. Use black line for the theoretical c.d.f.

1

ii. Use red diamonds for the experimental one.

iii. Include axes labels, title and legend.

e. Submit your MATLAB code with your comments on EACH LINE

Solution

%a. Generating the samples
%mean is 5, so distribute is exp( -x/5 )
samples = exprnd(5,100000,1); %returns 100000*1 array with sample numbers

%b. Plotting the histogram
mini = min( samples );
maxi = max( samples );
bins= 50;
h = hist(samples, bins);
xs = zeros( bins ,1);
for i=1:bins
    xs(i,1) = mini + ((i-1)*(maxi- mini))/bins;
end
figure;
bar( xs, h );
xlabel( \'bins\' );
ylabel( \'histogram\' );

%c. PDF
figure;
z = zeros( bins, 1);
theoriticalY = zeros( bins, 1 );
for i=1:size(z,1)
    z(i,1) = 0.2*(h(1,i)/max(h));
    theoriticalY(i,1) = 0.2*exp(-0.2*xs(i,1));
end
plot( xs, z, \'rd\');
hold on;
plot( xs, theoriticalY, \'b\');
xlabel( \'x\' );
ylabel( \'pdf of exp(-0.2x)\' );
title( \'Experimental and theoretical PDF of exp R.V.\');
legend( \'Experimental PDF\', \'Theoritical PDF\' );

%d. CDF
figure;
z = zeros( bins, 1);
theoriticalY = zeros( bins, 1 );
tillNow = 0;
for i=1:size(z,1)
    tillNow = tillNow + h(1,i);
    z(i,1) = (tillNow/100000);
    theoriticalY(i,1) = 1 - exp(-0.2*xs(i,1));
end
plot( xs, z, \'rd\');
hold on;
plot( xs, theoriticalY, \'b\');
xlabel( \'x\' );
ylabel( \'cdf of exp(-0.2x)\' );
title( \'Experimental and theoretical CDF of exp R.V.\');
legend( \'Experimental CDF\', \'Theoritical CDF\' );