Question

I need help integrating to find the PDF CDF in MATLAB. (a)

1. Generate six sequences of uniform random variables (length 10,000) in the range [-0.5,0.5]. Each element in the ith sequence can be considered to be one realization of the random variable X, for i- 1,2,..., 6. Note that these six uniform random variables are independent and identically distributed. Com- pute new random variables as follows: 6 i-1 This results in two new sequences, the elements of which can be thought of as 10,000 separate realizations of the random variables Y1 and Y2 (a) Experimentally determine the PDF and CDF of Y1 and Y2 by developing a normalized histogram and integrating. (It is recommended that you use about 20 to 30 intervals or "bins" for this work.)

Answer 1

Below is the Matlab Code

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

close all,
clear all,
clc,

a = -0.5;
b = 0.5;
x_1=[];
x_2=[];
x_3=[];
x_4=[];
x_5=[];
x_6=[];
Y_1=[];
y_2=[];
for i=1:10000
x_1(i) = (b-a)*rand + a;
x_2(i) = (b-a)*rand + a;
x_3(i) = (b-a)*rand + a;
x_4(i) = (b-a)*rand + a;
x_5(i) = (b-a)*rand + a;
x_6(i) = (b-a)*rand + a;
y_1(i) = x_1(i) + x_2(i) + x_3(i) + x_4(i) + x_5(i) + x_6(i);
y_2(i) = x_1(i) + (0.5*x_2(i)) + (0.8*x_3(i)) + (1.8*x_4(i)) + (0.3*x_5(i)) + (0.5*x_6(i));
end

subplot(3,2,1); plot(x_1); title('X1');
subplot(3,2,2); plot(x_2); title('X2');
subplot(3,2,3); plot(x_3); title('X3');
subplot(3,2,4); plot(x_4); title('X4');
subplot(3,2,5); plot(x_5); title('X5');
subplot(3,2,6); plot(x_6); title('X6');

figure,
subplot(2,1,1); hist(y_1,20); title('PDF - Y1');
subplot(2,1,2); hist(y_2,20); title('PDF - Y2');

figure,
subplot(2,1,1); cdfplot(y_1); title('CDF - Y1');
subplot(2,1,2); cdfplot(y_2); title('CDF - Y2');

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

X1 0.5 0.5 0.5 0.5 0 1… 〕00 3000 4000 5Ⅲ 6000 7000 8000 SOO 10000 1000 2000 3000 4000 5000 E000 7000 8000 9000 10000 XA 3 0.5 0.5 0.5 0.5 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 000 10000 1000 2… 3000 4000 5000 6000 7000 2010 9000 1 0 x5 0.5 0.5 0.5 0.5 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

PDF - Y1 0.5 2.5 PDF Y2 1200 200 0.5 0.5 2.5

CDF-Y1 0.8 0.6 0.5 2.5 0.5 0.5 2.5