files added

Author: vikrampande7

ECE514 Project Part 1 Instruction.pdf

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+clc;
+clear;
+close all;
+
+T1=100;
+T2=1000;
+T3=10000;
+
+rng(42);
+%For Normal Random Variables with Mean=2 and Variance=2
+figure(1) % To plot Routine Method
+
+e1 = exprnd((1/2),1,T1);
+subplot(2,2,1)
+h1=histogram(e1);
+me1 = mean(e1); ve1 = var(e1); se1 = std(e1);
+title("exp~(2)");
+xlabel("100 Samples (Routine Method)")
+ylabel("Frequency")
+%hold on;
+grid on;
+e2 = exprnd((1/2),1,T2);
+subplot(2,2,2)
+me2 = mean(e2); ve2 = var(e2); se2 = std(e2);
+h2=histogram(e2);
+grid on;
+title("exp~(2)")
+xlabel("1000 Samples (Routine Method)")
+ylabel("Frequency")
+e3= exprnd((1/2),1,T3);
+me3 = mean(e3); ve3 = var(e3); se3 = std(e3);
+subplot(2,2,3)
+h3=histogram(e3);
+grid on;
+title("exp~(2)")
+xlabel("10000 Samples (Routine Method)")
+ylabel("Frequency")
+
+
+
+%Rejection Method
+figure(2); % To plot Rejection Method
+X1=rejection_exponential(T1);
+mx1 = mean(X1); vx1 = var(X1); sx1 = std(X1);
+disp("m1_rej" + mean(X1));
+disp("v1_rej" + var(X1))
+disp("s1_rej" + std(X1))
+subplot(2,2,1)
+X1=histogram(X1);
+grid on;
+title("exp~(2)")
+xlabel("100 Samples (Rejection Method)")
+ylabel("Frequency")
+X2=rejection_exponential(T2);
+mx2 = mean(X2); vx2 = var(X2); sx2 = std(X2);
+disp("m3_rej" + mean(X2));
+disp("v3_rej" + var(X2))
+disp("s3_rej" + std(X2))
+subplot(2,2,2)
+X2=histogram(X2);
+grid on;
+title("exp~(2)")
+xlabel("1000 Samples (Rejection Method)")
+ylabel("Frequency")
+X3=rejection_exponential(T3);
+mx3 = mean(X3); vx3 = var(X3); sx3 = std(X3);
+subplot(2,2,3)
+X3=histogram(X3);
+grid on;
+title("exp~(2)");
+xlabel("10000 Samples (Rejection Method)")
+ylabel("Frequency")
+
+function [X] = rejection_exponential(N)
+c = 1;
+X = zeros(N,1); % Preallocate memory
+f=@(x)(2*exp(-2*x));
+for i = 1:N
+    accept = false;
+    while accept == false
+        u = rand();
+        v = 4.*rand();
+        if c*u <= f(v)
+           X(i) = v;
+           accept = true;
+        end
+    end
+end
+end

ECE514 Project Part 2 Instruction.pdf

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+clc;
+clear;
+close all;
+
+T1=100;
+T2=1000;
+T3=10000;
+
+rng(7);
+
+%For Normal Random Variables with Mean=2 and Variance=2
+figure(1) % To plot Routine Method
+
+n1=normrnd(2,sqrt(2),[T1,1]);
+subplot(2,2,1)
+h1=histogram(n1);
+title("N~(2,2), Mean= " + mean(n1) + " Var= " + var(n1) + " Std = " + std(n1));
+xlabel("100 Samples (Routine Method)")
+ylabel("Frequency")
+%hold on;
+grid on;
+n2= normrnd(2,sqrt(2),[T2,1]);
+subplot(2,2,2)
+h2=histogram(n2);
+grid on;
+title("N~(2,2), Mean= " + mean(n2) + " Var= " + var(n2)+ " Std = " + std(n2))
+xlabel("1000 Samples (Routine Method)")
+ylabel("Frequency")
+n3=normrnd(2,sqrt(2),[T3,1]);
+subplot(2,2,3)
+h3=histogram(n3);
+grid on;
+title("N~(2,2),Mean= " + mean(n3) + "Var= " + var(n3)+ " Std = " + std(n3))
+xlabel("10000 Samples (Routine Method)")
+ylabel("Frequency")
+
+
+
+%Rejection Method
+figure(2); % To plot Rejection Method
+X1=rejection_normal(T1);
+mx1 = mean(X1); vx1 = var(X1); sx1 = std(X1);
+disp("m1_rej" + mean(X1));
+disp("v1_rej" + var(X1))
+disp("s1_rej" + std(X1))
+subplot(2,2,1)
+X1=histogram(X1);
+grid on;
+title("N~(2,2),T = 100, m = "+ mx1 + " v = " +vx1+ " std = " + sx1)
+X2=rejection_normal(T2);
+mx2 = mean(X2); vx2 = var(X2); sx2 = std(X2);
+disp("m3_rej" + mean(X2));
+disp("v3_rej" + var(X2))
+disp("s3_rej" + std(X2))
+subplot(2,2,2)
+X2=histogram(X2);
+grid on;
+title("N~(2,2),T = 1000, m = "+ mx2 + " v = " +vx2+ " std = " + sx2)
+X3=rejection_normal(T3);
+mx3 = mean(X3); vx3 = var(X3); sx3 = std(X3);
+subplot(2,2,3)
+X3=histogram(X3);
+grid on;
+title("N~(2,2),T = 10000, m = "+ mx3 + " v = " +vx3+ " std = " + sx3);
+
+
+function [X] = rejection_normal(N)
+ %a = -4;
+ %b = 4;
+ f=@(x)exp(-((x-2).^2)/4)*(1/(sqrt(2*pi)));
+ cmax = 1/(2*sqrt(2*pi))*exp(-1/4);
+ %cmax = 1/(sqrt(2*2*pi));
+ X = zeros(1,N);
+ for i = 1:N
+     accept = false;
+     while accept == false
+         u = rand(); %fx
+         v = 2+2.*randn(); %gx
+         if cmax*u <= f(v)
+             X(i) = v;
+             accept = true;
+
+         end
+     end
+ end
+end
+

Report.docx

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+clc;
+clear;
+close all;
+
+T1=100;
+T2=1000;
+T3=10000;
+a = 2;
+b = 4;
+rng(42);
+
+%For Uniform Random Variables with a=2 and b=4
+figure(1) % To plot Routine Method
+u1 = a + (b-a)*rand(1,T1);
+subplot(2,2,1)
+disp("m1 " + mean(u1))
+disp("v1 " + var(u1))
+disp("s1 " + std(u1))
+h1=histogram(u1);
+title("U~(2,4), Mean= " + mean(u1) + " Var= " + var(u1)+ " Std = " + std(u1));
+xlabel("100 Samples (Routine Method)")
+ylabel("Frequency")
+%hold on;
+grid on;
+u2 = a + (b-a)*rand(1,T2);
+subplot(2,2,2)
+disp("mean2: " + mean(u2))
+disp("var2: " + var(u2))
+disp("s3 " + std(u2))
+h2=histogram(u2);
+grid on;
+title("U~(2,4),Mean= " + mean(u2) + " Var= " + var(u2) + " Std = " + std(u2))
+xlabel("1000 Samples (Routine Method)")
+ylabel("Frequency")
+u3 = a + (b-a)*rand(1,T3);
+subplot(2,2,3)
+h3=histogram(u3);
+disp("m3 " +mean(u3));
+disp("v3 " + var(u3))
+disp("s3 " + std(u3))
+grid on;
+title("U~(2,4),Mean= " + mean(u3) + "Var= " + var(u3)+ " Std = " + std(u3))
+xlabel("10000 Samples (Routine Method)")
+ylabel("Frequency")
+
+
+
+%Rejection Method
+figure(2); % To plot Rejection Method
+X1=rejection_uniform(T1);
+subplot(2,2,1)
+mx1 = mean(X1); vx1 = var(X1); sx1 = std(X1);
+X1=histogram(X1);
+grid on;
+title("U~(2,4),T = 100, m = "+ mx1 + " v = " +vx1+ " std = " + sx1)
+X2=rejection_uniform(T2);
+mx2 = mean(X2); vx2 = var(X2); sx2 = std(X2);
+subplot(2,2,2)
+X2=histogram(X2);
+grid on;
+title("U~(2,4),T = 1000, m = "+ mx2 + " v = " +vx2+ " std = " + sx2)
+X3=rejection_uniform(T3);
+mx3 = mean(X3); vx3 = var(X3); sx3 = std(X3);
+subplot(2,2,3)
+disp("X3" + mean(X3))
+X3=histogram(X3);
+grid on;
+title("U~(2,4),T = 10000, m = "+ mx3 + " v = " +vx3+ " std = " + sx3)
+
+
+function [X] = rejection_uniform(N)
+c = 0.5;
+X = zeros(N,1); % Preallocate memory
+for i = 1:N
+    accept = false;
+    while accept == false
+        u = rand();
+        v = unifrnd(2,4);
+        if c*u <= 0.5
+           X(i) = v;
+           accept = true;
+        end
+    end
+end
+ 
+end

Results/Problem 1/exponential_reject.png

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+clc;
+clear;
+close all;
+
+T1 = 100;
+T2 = 1000;
+T3 = 10000;
+lambda = 2;
+
+% For 100 samples
+for i = 1:length(T1)
+    gx = zeros(T1(i),1);
+    n = 1;
+    while n <= T1(i)
+        fx = exprnd(lambda,T1(i),1);
+        gx(n) = sum(fx)/T1(i);
+        n = n + 1;
+    end
+    figure(1);
+    subplot(1,1,i)
+    histogram(gx)
+    title("TRANSFORMED EXPONENTIAL RANDOM VARIABLE", T1(i))
+    xlabel("100 Samples");
+    ylabel("Frequency");
+end
+
+% 1000 Samples
+for i = 1:length(T2)
+    gx = zeros(T2(i),1);
+    n = 1;
+    while n <= T2(i)
+        fx = exprnd(lambda,T2(i),1);
+        gx(n) = sum(fx)/T2(i);
+        n = n + 1;
+    end
+    figure(2);
+    subplot(1,1,i)
+    histogram(gx)
+    title("TRANSFORMED EXPONENTIAL RANDOM VARIABLE", T2(i))
+    xlabel("1000 Samples");
+    ylabel("Frequency");
+end
+
+% 10000 samples
+for i = 1:length(T3)
+    gx = zeros(T3(i),1);
+    n = 1;
+    while n <= T3(i)
+        fx = exprnd(lambda,T3(i),1);
+        gx(n) = sum(fx)/T3(i);
+        n = n + 1;
+    end
+    figure(3);
+    subplot(1,1,i)
+    histogram(gx)
+    title("TRANSFORMED EXPONENTIAL RANDOM VARIABLE", T3(i))
+    xlabel("10000 Samples");
+    ylabel("Frequency");
+end

Results/Problem 1/exponential_routine.png

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+clc;
+clear;
+close all;
+
+T1 = 100;
+T2 = 1000;
+T3 = 10000;
+lambda = 2;
+
+% For 100 samples
+for i = 1:length(T1)
+    gx = zeros(T1(i),1);
+    n = 1;
+    while n <= T1(i)
+        fx = normrnd(2,2,[T1,1]);
+        gx(n) = sum(fx)/T1(i);
+        n = n + 1;
+    end
+    figure(1);
+    subplot(1,1,i)
+    histogram(gx)
+    title("TRANSFORMED NORMAL RANDOM VARIABLE", T1(i))
+    xlabel("100 Samples");
+    ylabel("Frequency");
+end
+
+% 1000 Samples
+for i = 1:length(T2)
+    gx = zeros(T2(i),1);
+    n = 1;
+    while n <= T2(i)
+        fx = normrnd(2,2,[T2,1]);
+        gx(n) = sum(fx)/T2(i);
+        n = n + 1;
+    end
+    figure(2);
+    subplot(1,1,i)
+    histogram(gx)
+    title("TRANSFORMED NORMAL RANDOM VARIABLE", T2(i))
+    xlabel("1000 Samples");
+    ylabel("Frequency");
+end
+
+% 10000 samples
+for i = 1:length(T3)
+    gx = zeros(T3(i),1);
+    n = 1;
+    while n <= T3(i)
+        fx = normrnd(2,2,[T3,1]);
+        gx(n) = sum(fx)/T3(i);
+        n = n + 1;
+    end
+    figure(3);
+    subplot(1,1,i)
+    histogram(gx)
+    title("TRANSFORMED NORMAL RANDOM VARIABLE", T3(i))
+    xlabel("10000 Samples");
+    ylabel("Frequency");
+end