Stochastic assignment Stochastic Systems Assignment Title of Assignment Table 3.1+3.2 implementation on Matlab Submitted to: Sir Dr. Salman Submitted by: Sikandar Javed Reg # 7326 Ms-83/Electrical(control) Engineering Page 1
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Stochastic Systems Assignment
Title of Assignment
Table 3.1+3.2 implementation on Matlab
Submitted to:
Sir Dr. Salman
Submitted by:
Sikandar Javed
Reg # 7326
Ms-83/Electrical(control) Engineering
College of Engineering & Mechanical Engineering, NUST
Page 1
Bernoulli Random Variable:
p=0:1p_0=1-p %pmf=p for sucees=0p_1=p %pmf=p for sucees=1Mean=mean(p_1)Var=var(p_1)%in tableE=pVAR=p.*(1-p)plot(p,p_1)
Page 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Binomial (pmf)N=50P=0.1
% binomial random variable %clcclear alln=50; % number of trialsp=0.1 %probability of successfor k=0:n % probability mass function pmf=((factorial(n))/((factorial(k))*(factorial(n-k))))*(p^k)*((1-p)^(n-k) ) hold on stem(k,pmf)end%according to table 3.1MEAN=mean(pmf) VARiance=var(pmf) E=n*p %farmulaVar=n*p*(1-p)%farmula
Page 3
0 5 10 15 20 25 30 35 40 45 500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Binomial (pmf)N=50P=0.5% binomial random variable %clcclear alln=50; % number of trialsp=0.5 %probability of successfor k=0:n % probability mass function pmf=((factorial(n))/((factorial(k))*(factorial(n-k))))*(p^k)*((1-p)^(n-k) ) hold on stem(k,pmf)end%according to table 3.1MEAN=mean(pmf) VARiance=var(pmf) E=n*p %farmulaVar=n*p*(1-p)%farmula
Page 4
0 5 10 15 20 25 30 35 40 45 500
0.02
0.04
0.06
0.08
0.1
0.12
Geometric random variable(pmf)
n=10;p=0.2for k=1:n pmf=(p)*((1-p)^(k-1) ) %P[M=k]=(1-p)^(k-1) *P hold on stem(k,pmf)end
Page 5
%according to table 3.1MEAN=mean(pmf) VARiance=var(pmf) E=(1-p)/p % expected value farmulaVar=(1-p)/p^2 % variance value farmula
1 2 3 4 5 6 7 8 9 100
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
%
For CDF:
n=15;p=0.3for k=1:n pmf=(p)*((1-p)^(k-1) ) %P[M=k]=(1-p)^(k-1) *P T=@(k) pmf; CDF=integral(T,1,15,'ArrayValued',true) hold on stem(k,cdf) hold offendMEAN=mean(pmf) VARiance=var(pmf) E=(1-p)/p Var=(1-p)/p^2
Page 6
0 5 10 150
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Negative random variable(pmf)
nagitive binomial random variable
n=20;r=2;p=0.5for k=r:r+npmf=((factorial(k-1))/((factorial(r-1))*(factorial(k-r))))*((p^r)*((1-p)^(k-r))) hold on stem(k,pmf)end%according to table 3.1MEAN=mean(pmf)
Page 7
VARiance=var(pmf) E=r/p %farmulaVar=(r*(1-p))/p^2 %farmula
2 4 6 8 10 12 14 16 18 20 220
0.05
0.1
0.15
0.2
0.25
For CDF:
%USING nEGATIVE bINOMIAL r.vn=10;r=1;p=0.5for k=r:r+npmf=((factorial(k-1))/((factorial(r-1))*(factorial(k-r))))*((p^r)*((1-p)^(k-r)))T=@(k) pmf; CDF=integral(T,1,10,'ArrayValued',true)
hold on stem(k,CDF) hold offendMEAN=mean(pmf) % according to commandVARiance=var(pmf) %acc to Matlab command E=r/p Var=(r*(1-p))/p^2
Page 8
Poisson random variable(pmf)n=20;p=0.3e=2.713alfa=0.8for k=0:n pmf=(alfa^k/factorial(k))*e^-alfa hold on stem(k,pmf) hold offend% according to table 3.1MEAN=mean(pmf) VARiance=var(pmf) E=alfa %farmulaVar=alfa %farmula
0 2 4 6 8 10 12 14 16 18 200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
FOR CDF:
At Alfa=0.75
n=24;
Page 9
p=0.5e=2.713alfa=0.75for k=0:n pmf=(alfa^k/factorial(k))*e^-alfa T=@(k) pmf; CDF=integral(T,1,24,'ArrayValued',true)
hold on stem(k,CDF) hold offendMEAN=mean(pmf) % according to commandVARiance=var(pmf) %acc to Matlab command E=alfa Var=alfa
Uniform random variable
a=1b=10 for x=a:b cdf=(x-a)./(b-a) ; pdf=polyder(cdf) hold on plot(x,cdf,'b')end%according to table 3.2MEAN=mean(cdf) VARiance=var(cdf) E=(a+b)/2 Var=(b-a)^2/12
Page 10
1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
For CDF:
a=1b=10 for x=a:b cdf=(x-a)./(b-a) ; pdf=polyder(cdf) %as pdf=differentiation(cdf) T=@(x) pdf; CDF=integral(T,1,10,'ArrayValued',true)
hold on subplot(2,1,1) plot(x,cdf,'b') subplot(2,1,2) plot(x,pdf,'b') hold offendMEAN=mean(cdf) % according to command
Page 11
VARiance=var(cdf) %acc to Matlab command E=(a+b)/2Var=(b-a)^2/12
7) For Exponential Random Variable:
n=15p=0.4alfa=n*pT=10 %T=time in secondslameda=alfa/Tfor x=0:n pdf=lameda*exp(-lameda.*x) hold on plot(x,pdf)endMEAN=mean(pdf) % according to commandVARiance=var(pdf) %acc to Matlab command E=1/lameda %farmulaVar=1/lameda^2 %farmula
0 5 10 15 20 250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
FOR CDF:
n=15p=0.4alfa=n*p
Page 12
T=10 %T=time in secondslameda=alfa/Tfor x=0:n pdf=lameda*exp(-lameda.*x) T=@(x) pdf; CDF=integral(T,0,15,'ArrayValued',true)
hold on plot(x,pdf) hold offendMEAN=mean(pdf) % according to commandVARiance=var(pdf) %acc to Matlab command E=1/lamedaVar=1/lameda^2
For Guassian Random Variable
sigma=0.5m=5;for x=0:10 pdf=(exp(-(x-m)^2)/(2*sigma^2))./((sqrt(2*pi))*sigma) hold on plot(x,pdf) % according to table 3.1MEAN=mean(pdf) VARiance=var(pdf) E=m Var=sigma.^2
For Sigma=0.4;0.5;0.6
Page 13
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
FOR CDF:
%For Guassian Random Variablesigma=0.5m=5;for x=0:10 pdf=(exp(-(x-m)^2)/(2*sigma^2))./((sqrt(2*pi))*sigma)T=@(x) pdf; CDF=integral(T,0,10,'ArrayValued',true)
hold on plot(x,pdf) hold offendMEAN=mean(pdf) % according to commandVARiance=var(pdf) %acc to Matlab command E=mVar=sigma.^2
9. For Gamma Random Variable:
Page 14
FOR PDF:
alfa=0.5m=0;lameda=1;z=0.5;for x=0:10 fun=@(x) x.^(z-1).*exp(-x); T=integral(fun,0,10) %T=gamma function pdf=(lameda.*(lameda.*x).^(alfa-1)).*(exp(-lameda.*x))./T hold on plot(x,pdf) hold offendMEAN=mean(pdf) % according to commandVARiance=var(pdf) %acc to Matlab command E=alfa/lamedaVar=alfa/lameda.^2
For Alfa=0.5;1;2 and Lameda=1:
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
FOR CDF:
Page 15
alfa=0.5m=0;lameda=1;z=0.5;for x=0:10 fun=@(x) x.^(z-1).*exp(-x); T=integral(fun,0,10) %T=gamma function pdf=(lameda.*(lameda.*x).^(alfa-1)).*(exp(-lameda.*x))./TT=@(x) pdf;CDF=integral(T,0,10,'ArrayValued',true)
hold on plot(x,pdf) hold offendMEAN=mean(pdf) % according to commandVARiance=var(pdf) %acc to Matlab command E=alfa/lamedaVar=alfa/lameda.^2
10. For Rayleigh Random Variable:
FOR PDF:%For Guassian Random Variablesigma=0.5alfa=1;for x=0:10 pdf=(x./alfa.^2).*(exp((-x^2)./(2*sigma.^2))) hold on plot(x,pdf) hold offendMEAN=mean(pdf) % according to commandVARiance=var(pdf) %acc to Matlab command E=alfa.*(sqrt(pi./2))Var=(2-pi/2).*alfa.^2
For Alfa=1;0.7;0.5
Page 16
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
FOR CDF:
%For Guassian Random Variablesigma=0.5alfa=1;for x=0:10 pdf=(x./alfa.^2).*(exp((-x^2)./(2*sigma.^2)))T=@(x) pdf; CDF=integral(T,0,10,'ArrayValued',true)
hold on plot(x,pdf) hold offendMEAN=mean(pdf) % according to commandVARiance=var(pdf) %acc to Matlab command E=alfa.*(sqrt(pi./2))Var=(2-pi/2).*alfa.^2
Page 17
11. For Cauchy Random Variable:
For PDF:
%For Cauchhy Random Variablealfa=5;for x=0:10 pdf=(alfa./pi)./((x.^2)+(alfa.^2)) hold on plot(x,pdf) hold offendMEAN=mean(pdf) % according to commandVARiance=var(pdf) %acc to Matlab command
0 1 2 3 4 5 6 7 8 9 100.01
0.02
0.03
0.04
0.05
0.06
0.07
For CDF:
%For Cauchhy Random Variable
Page 18
alfa=5;for x=0:10 pdf=(alfa./pi)./((x.^2)+(alfa.^2))T=@(x) pdf; CDF=integral(T,0,10,'ArrayValued',true) hold on plot(x,pdf) hold offendMEAN=mean(pdf) % according to commandVARiance=var(pdf) %acc to Matlab command
12.For Laplacian Random Variable:
FOR PDF:
%For Cauchhy Random Variablealfa=5;for x=0:10 pdf=(alfa./2).*(exp(-alfa.*x)) hold on plot(x,pdf) hold offendMEAN=mean(pdf) % according to commandVARiance=var(pdf) %acc to Matlab commandE=0 %acc to table 3.2Var=2/alfa.^2 %acc to table 3.2
Page 19
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
FOR CDF:
%For Cauchhy Random Variablealfa=5;for x=0:10 pdf=(alfa./2).*(exp(-alfa.*x)) T=@(x) pdf; CDF=integral(T,0,10,'ArrayValued',true)
hold on plot(x,pdf) hold offendMEAN=mean(pdf) % according to commandVARiance=var(pdf) %acc to Matlab commandE=0 %acc to table 3.2Var=2/alfa.^2 %acc to table 3.2
Page 20
Related Documents
