Final Matlab

2.2 EXERCISES WITH MATLAB 2.2.1 Standard Normal Distribution Load dat1_3 containing the random sequences x and y of two bivariate normal distributed randomvariables. Transform the random samples such that they obey a standard normal distribution.Make sure that your transformation is correct by showing the theoretical and the via density estimated density of the transformed random sequence x in a diagram. Program clear all; close all; load dat1_3 Meanx = sum(x) / 1000; Meany = sum(y) / 1000; X = (x - Meanx)/ cov(x); [prob,x_cord] = density(X); z = -4 + 8/1000 : 8/1000 : 4; Y = (1 / sqrt(2 * pi)) * exp ( (- z .^ 2)/2); x_cord2 = -4 + 8/1000 : 8/1000 : 4; plot(x_cord,prob,'--',x_cord2,Y) grid on title('STANDARD NORMAL DISTRIBUTION(DENSITY PLOT)') legend('Practical Density','Theorectical Density','tr') xlabel('X') ylabel ('fx(x)')

Density function function [prob,x_cord] = density(x) [ten_div x_cord] = hist(x); interval = x_cord(2) - x_cord(1); l=length(x); prob = ten_div * (1 / (interval*l)); end

Output S T A N D A R D N O R M A LD IS T R IB U T IO N (D E N S IT Y P L O T ) 0 .4 P ra c tic a lD e n s ity T h e o re c tic a lD e n s ity

0 .3 5 0 .3

fx(x)

0 .2 5 0 .2 0 .1 5 0 .1 0 .0 5 0 -4

-3

-2

-1

0 X

1

2

3

4

2.2.2 Exponential Distribution Load the on [0, 1] uniformly distributed random sequence from dat1_1 and transform it as explained in section 2.1.2 with α = 1/ 2 . Calculate the mean value and variance of the transformedsequence and compare the results with the theoretical values from section 2.1.2. Show the in section2.1.2 calculated and via density estimated density function of the transformed random sequence in a figure. program clear all; close all; load dat1_1; X=abc; Y=-2*log(X); mean_p = mean(Y); variance_p = var(Y); mean_theory=2;

variance_theory=4; [pd_p,k]=density(Y); Y_theory=.001:.001:16; pd_Y_theory=0.5*exp(-0.5*Y_theory); plot(k,pd_p,'-',Y_theory,pd_Y_theory,'--','linewidth',2) grid on title('EXPONENTIAL DISTRIBUTION (DENSITY PLOT)') legend('Practical Density','Theoretical Density','tr') xlabel('X') ylabel ('f(x)'

Output Mean practical= 2.0275 Variance practical= 4.4518 Mean theoretical=2.00 Variance theoretical=4.00

EXPONENTIAL DISTRIBUTION (DENSITY PLOT) 0.5 Practical Density Theoretical Density

0.45 0.4 0.35

f(x)

0.3 0.25 0.2 0.15 0.1 0.05 0

0

2

4

6

8 X

10

12

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16

2.2.3 Sum of random numbers Generate two on [0, 1] uniformly distributed random sequences of the length 1000 after setting the initial value to 0. Save both random sequences in dat2_1. Then add the two sequences element-by-element. Calculate the mean value and variance of the sum and compare the results withthe theoretical values from section 2.1.3. Display the in section 2.1.3 calculated and via densityestimated density function in a diagram. program clc clear all; rand('state',0);

dat2_1=rand(1000,2); def=dat2_1; x=sum(dat2_1,2); mean_x=mean(x); variance_x=var(x); [x_pra_density,x_cord]=density(x); %theory n=1; for i=.001:.001:2 x_theory(n)=i; if n<=1000 x_theory(n)=x_theory(n); else x_theory(n)=2-x_theory(n); end x_cord_theory(n)=i; n=n+1; end plot(x_cord,x_pra_density,'-',x_cord_theory,x_theory,' .','linewidth',2); grid on; title('SUM OF RANDOM VARIABLES (DENSITY PLOT)'); legend('Practical Density','Theoretical Density'); xlabel('x'); ylabel ('fx(x)');

Output Mean practical= 1.0014 Variance practical= 0.1766 Mean theoretical=1 Variance theoretical=0.166

SUM OF RANDOM VARIABLES (DENSITY PLOT) 1 Practical Density Theoretical Density

0.9 0.8 0.7

fx(x)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1 x

1.2

1.4

1.6

1.8

2

2.2.4 Product of random numbers Multiply both on [0, 1] uniformly distributed random sequences from dat2_1 element-byelement.Calculate the mean value and variance of the product and compare the results with the theoretical values derived in section 2.1.4. Depict the in section 2.1.4 calculated and via density estimated density function in a figure. Program clear all; clc; load dat2_1 def; x=def(:,1); y=def(:,2); z=x.*y mean_z=mean(z); var_z=cov(z); [prob,x_cord]=density(z); n=1;

for i=0:1/1000:1 y_cord2(n)=-log(i); n=n+1; end x_cord2=0:1/1000:1; plot(x_cord,prob,'--',x_cord2,y_cord2,’line width’,2); grid on; title('Product of two random variables'); legend('Practical Density','Theoretical Density','tr'); xlabel('x'); ylabel ('fx(x)');

Output Mean practical=0.254 Variance practical=0.053 Mean theoretical=0.25 Variance theoretical=0.05 Product of tworandomvariables 7 Practical Density Theoretical Density

6

5

fx(x)

4

3

2

1

0 0

0.1

0.2

0.3

2.2.5 χ2-distribution

0.4

0.5 x

0.6

0.7

0.8

0.9

1

Generate four standard normal distributed random sequences of the length 1000, where before theinitial value has to be set to 0. Save the data in dat2_2. Determine the sum of squares of the four random sequences element-by-element. Calculate the mean value and variance of the sum and compare the results with the theoretical values obtained in section 2.1.5. Present the in section2.1.5 calculated and via density estimated density function in a diagram. Program clear all; close all; randn('state',0); % standard normal distributed randon sequences x1 = randn(1,1000); x2 = randn(1,1000); x3 = randn(1,1000); x4 = randn(1,1000); save dat2_2 x1 x2 x3 x4 x11 = x1.*x1; x22 = x2.*x2; x33 = x3.*x3; x44 = x4.*x4; % X = x1^2 + x2^2 + x3^2 + x4^2 x = x11 + x22 + x33 + x44; Mean_x = mean(x); Variance_x = var(x); mean_theory=4; variance_theory=8; [x_pra_density,x_cord] = density(x); x_cord_theory = min(x_cord):0.01:max(x_cord); % fx(x) = (x^((n/2)-1) * exp(-x/2))/(2^n/2 * ((n/2)-1)!) pd_theory1 = x_cord_theory.* exp(-(x_cord_theory/2)); pd_theory = (1/4) * pd_theory1; plot(x_cord_theory,pd_theory,'-',x_cord,x_pra_density,'--','linewidth',2) grid on title('CHI-SQUARE DISRIBUTION(DENSITY PLOT)') legend('Theoretical','Practical','tr') xlabel('x') ylabel('fx(x)')

Output

Mean practical=3.9874 Variance practical=7.5348 Mean theoretical=4 Variance theoretical=8 CHI-SQUARE DISRIBUTION(DENSITY PLOT) 0.2 Theoretical Practical

0.18 0.16 0.14

fx(x)

0.12 0.1 0.08 0.06 0.04 0.02 0

0

2

4

6

8

10 x

12

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18

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2.2.6 Normal distribution from uniform distribution Now use again dat2_1 containing two on [0, 1] uniformly distributed random sequences. Build y1 as in section 2.1.6 explained. Calculate the mean value and variance of y1 and compare the results with the theoretical values determined in section 2.1.6. Show the in section 2.1.6 calculatedand via density estimated density function in a figure. Program clear all; close all; load dat2_1 def; x=def(:,1);

y=def(:,2); % Generating the sequence Y1 Y1 = sqrt(-2*log(x)).*sin(2*pi*y); mean_p_Y1 = mean(Y1); var_p_Y1 = var(Y1); % Density function [pd_pra_Y1,x_cord_pra] = density(Y1); x_cord_theory = min(x_cord_pra) : 0.01 :max(x_cord_pra); pd_theory_Y1= 1/sqrt(2*pi)*exp(-x_cord_theory.^2/2); plot(x_cord_theory,pd_theory_Y1,'--',x_cord_pra,pd_pra_Y1,'-','linewidth',2); grid on title('NORMAL DISTRIBUTION FROM UNIFORM DISTRIBUTION(DENSITY PLOT)') legend('TheoreticaL','Practical','tr') xlabel('x values') ylabel ('density')

Output Mean practical=0.0371 Variance practical=1.0174 Mean theoretical=0 Variance theoretical=1 NORMAL DISTRIBUTIONFROMUNIFORMDISTRIBUTION(DENSITY PLOT) 0.4 TheoreticaL Practical

0.35 0.3

density

0.25 0.2 0.15 0.1 0.05 0 -3

-2

-1

0 x values

1

2

3