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Daniel Cujba
2024-08-31 12:07:21 +03:00
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Anul 2/Semestrul 1/Probabilitati si statistica/Main.m
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#A=[1 2 3; 4 5 6; 7 8 9];
#x=1:2:10;
#A(1:2,2:3);
#x+2;
#3*x;
#x.^2;
#B=[5 4 3; 8 2 1; 1 5 5];
#A*B;
#B*A;
A=[1 0 -2; 2 1 3; 0 1 0];
B=[2 1 1; 1 0 -1;1 1 0];
C=A-B
D=A*B
E=A.*B
x=0:0.01:3;
plot(x, x.^5/10,"k:",x,x.*sin(x),"r--",x,cos(x),"b")
Anul 2/Semestrul 1/Probabilitati si statistica/fun1.m
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## Copyright (C) 2023 danie
##
## This program is free software: you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program. If not, see <https://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {} {@var{retval} =} fun1 (@var{input1}, @var{input2})
##
## @seealso{}
## @end deftypefn
## Author: danie <danie@DANIELCUJBA>
## Created: 2023-10-09
function y = fun1 (x)
y=x^2-x
endfunction
Anul 2/Semestrul 1/Probabilitati si statistica/lab2_2.m
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# We are solving ex 2
# We plot the pdf and cdf of the binomial
n = input("Give the number of trials n=");
p = input("Give the probability of success p=");
x=0:1:n;
px=binopdf(x,n,p);
plot(x,px,'*r');
hold on
xx=0:0.0001:n;
cx=binocdf(xx,n,p);
plot(xx,cx,'g');
Anul 2/Semestrul 1/Probabilitati si statistica/lab3_1.m
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n = input("Give the degrees of freedom n=");
#a
chi2cdf(0,n);
1-chi2cdf(0,n);
#b
chi2cdf(1,n)-chi2cdf(-1,n);
1-(chi2cdf(1,n)-chi2cdf(-1,n));
#c+d
a = input("Give a number between 0 and 1 a=");
b = input("Give a number between 0 and 1 b=");
chi2inv(a,n)
chi2inv(1-b,n)
Anul 2/Semestrul 1/Probabilitati si statistica/lab_2_app.m
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# X ~ Bino(3,0.5)
# We are solving Application from lab 2
n=3 # 3 coin flips
p=0.5 # coin flips <=> 0.5 probability
x=0:1:n;
px=binopdf(x,n,p);
# ( 0 1 2 3 )
# X ( 1/8 3/8 3/8 1/8)
#
plot(x,px,"+");
hold on
# P(X=0) = binopdf(0,3,0.5) =1/8
p1=binopdf(0,3,0.5);
printf('P(X=0)=%1.6f\n',p1)
xx=0:0.01:n;
# P(X!=0) = 1 - binopdf(1,3,0.5) =5/8
p2=1-binopdf(1,3,0.5); # PROBABILITY OF THE COMPLAMENTARY EVENT
printf('P(X!=0)=%1.6f\n',p2)
#P(X<=2) = binocdf(2,3,0.5) = 7/8
p3=binocdf(2,3,0.5);
printf('P(X<=2)=%1.6f\n',p3)
#P(X<2) = binocdf(1,3,0.5) = 1/2 P(X<2)=P(X<=1)
p4=binocdf(1,3,0.5)
printf('P(X<2)=%1.6f\n',p4)
# P(X>=1) = 1-P(X<1) = 1 - binocdf(0,3,0.5)
p5=1 - binocdf(0,3,0.5); # PROBABILITY OF THE COMPLAMENTARY EVENT
printf('P(X>=1)=%1.6f\n',p5)
# P(X>1) = 1 - P(X<=1) - binocdf(1,3,0.5)
p6=1 - binocdf(1,3,0.5); # PROBABILITY OF THE COMPLAMENTARY EVENT
printf('P(X>1)=%1.6f\n',p6)
N=input("Give the number of simulations N=");
U=rand(3,N); # Generate N simulations of 3 coin tosses
Y=(U>0.5); # Transforms decimal in True for Heads(>0.5) or False for Tails
X=sum(Y); # Sums up the collums
clf # Clear the figure
hist(X); # Plots X
Anul 2/Semestrul 1/Probabilitati si statistica/lab_3_2.m
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p = input("Give a probability between 0 and 1 p=");
for n = 1:3:100
x=0:n;
y=binopdf(x,n,p);
plot(x,y)
pause(0.5)
endfor
Anul 2/Semestrul 1/Probabilitati si statistica/lab_3_3.m
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p = input("Give a probability between 0 and 0.05 p=");
n = input("Give a number greater than 30 n=");
l=n*p;
for n = 1:3:100
x=0:n;
y=binopdf(x,n,p);
plot(x,y)
pause(0.5)
endfor
Anul 2/Semestrul 1/Probabilitati si statistica/lab_5_B1.m
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X=[7 7 4 5 9 9 4 12 8 1 8 7 3 13 2 1 17 7 12 5 6 2 1 13 14 10 2 4 9 11 3 5 12 6 10 7];
n=length(X);
#oneminusalpha = input("Confidence level: (between 0 and 1)");
oneminusalpha = 0.95;
alpha = 1-oneminusalpha;
sigma=5;
m1=mean(X)-(sigma/sqrt(n))*norminv(1-alpha/2, 0, 1);
m2= mean(X)-(sigma/sqrt(n))*norminv(alpha/2, 0, 1);
printf("Confidence interval for the theoretical mean when sigma is known is (%4.3f, %4.3f)\n", m1,m2);
m3=mean(X)-(std(X)/sqrt(n))*tinv(1-alpha/2,n-1);
m4= mean(X)-(std(X)/sqrt(n))*tinv(alpha/2,n-1);
printf("Confidence interval for the theoretical mean when sigma is unknown is (%4.3f, %4.3f)\n", m3,m4);
v1=((n-1)*var(X))/(chi2inv(1-alpha/2, n-1));
v2=((n-1)*var(X))/(chi2inv(alpha/2,n-1));
printf("Confidence interval for the variance is (%4.3f, %4.3f)\n", v1,v2);
s1=sqrt(v1);
s2=sqrt(v2);
printf("Confidence interval for the theoretical standard variance is (%4.3f, %4.3f)\n", s1,s2);
Anul 2/Semestrul 1/Probabilitati si statistica/lab_5_B2.m
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x1=[22.4 21.7 24.5 23.4 21.6 23.3 22.4 21.6 24.8 20.0];
x2=[17.7 14.8 19.6 19.6 12.1 14.8 15.4 12.6 14.0 12.2];
n1=length(x1);
n2=length(x2);
#oneminusalpha = input("Confidence level: (between 0 and 1)");
oneminusalpha = 0.95;
alpha = 1-oneminusalpha;
a1= (mean(x1) - mean(x2) - tinv(1-alpha/2,n1+n2-2)*sqrt(((n1-1)*var(x1)+(n2-2)*var(x2))/(n1+n2-2))*sqrt(1/n1+1/n2));
a2=(mean(x1) - mean(x2) + tinv(1-alpha/2,n1+n2-2)*sqrt(((n1-1)*var(x1)+(n2-2)*var(x2))/(n1+n2-2))*sqrt(1/n1+1/n2));
printf("The Confidence interval between population means when sigma1=sigma2 is (%4.3f, %4.3f)\n", a1,a2);
Anul 2/Semestrul 1/Probabilitati si statistica/lab_6_1.m
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X = [7 7 4 5 9 9 4 12 8 1 8 7 3 13 2 1 17 7 12 5 6 2 1 13 14 10 2 4 9 11 3 5 12 6 10 7];
n=length(X);
#oneminusalpha = input("Confidence level: (between 0 and 1)");
oneminusalpha = 0.95;
alpha = 1-oneminusalpha;
sigma=5;
#The null hypothesis is: H0: u=8.5 (it goes togheter with u>8.5, the standard is meet)
#The alternative hypothesis is: H1: u<8.5 (the standard is not meet)
#Left tail test for u when sigma is known
printf("This is a left tail test for the mean when sigma is known\n\n");
#n0 = input("Test value n0=");
n0=8.5;
[H, PVAL, CI, ZVALUE] = ztest(X,n0,sigma,'alpha',alpha,'tail','left');
z_alpha = norminv(alpha, 0, 1);
RR = [-inf z_alpha];
printf("The value of h is: %d\n\n",H);
if H==1
printf("The null hypothesis is rejected\n\n");
printf("The data suggests that the standard is not meet\n\n");
else
printf("The null hypothesis is not rejected\n\n");
printf("The data suggests that the standard is meet\n\n");
endif
printf("The regection region is: (%4.3f, %4.3f)\n",RR);
printf("The observed value of the test statistic is: %4.3f\n",ZVALUE );
printf("The Pvalue of the test is: %4.3f\n",PVAL);
#b use ttest
#2 a use vartest2
#2 b use ttest2
Anul 2/Semestrul 1/Probabilitati si statistica/test.m
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clear
X=[46 37 39 48 47 44 35 31 44 37];
Y=[35 33 31 34 34 30 27 32 31 31];
n1=length(X);
n2=length(Y);
alpha=0.05; # significance level
#The null hypothesis is: H0: vx!=vy (the standard is meet) (the variances differ)
#The alternative hypothesis is: H1: vx=vy (the standard is not meet) (the variances are the same)
#Both tail test when sigma is unknown
[H, PVAL, CI, STATS] = vartest2(X,Y,'alpha',alpha,'tail','both');
if H==1
printf("The null hypothesis is rejected\n\n");
printf("The data suggests that the standard is not meet\n\n");
printf("The variances seem to be the same");
else
printf("The null hypothesis is not rejected\n\n");
printf("The data suggests that the standard is meet\n\n");
printf("The variances seem to differ\n\n");
endif
z_alpha = norminv(alpha/2, 0, 1); # calculating tt of alpha/2
RR = [-inf z_alpha]; # first part of the regection region
z_alpha2 = norminv(1-alpha/2, 0, 1); # calculating tt of 1-alpha/2
RR2 = [z_alpha2 +inf]; # second part of the regection region
printf("The regection region is: (%4.3f, %4.3f) reunited with (%4.3f, %4.3f)\n",RR,RR2);
printf("The observed value of the test statistic is: %4.3f\n", STATS.fstat );
printf("The Pvalue of the test is: %4.3f\n\n",PVAL);
oneminualpha=0.95; # confidence level
a1= (mean(X) - mean(Y) - tinv(1-alpha/2,n1+n2-2)*sqrt(((n1-1)*var(X)+(n2-2)*var(Y))/(n1+n2-2))*sqrt(1/n1+1/n2));
a2=(mean(X) - mean(Y) + tinv(1-alpha/2,n1+n2-2)*sqrt(((n1-1)*var(X)+(n2-2)*var(Y))/(n1+n2-2))*sqrt(1/n1+1/n2));
printf("The Confidence interval between population means when sigma1=sigma2, where sigma and sigma2 are unknown, is (%4.3f, %4.3f)\n", a1,a2);
Anul 2/Semestrul 1/Probabilitati si statistica/test_14.m
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X=[4.6 0.7 4.2 1.9 4.8 6.1 4.7 5.5 5.4];
Y=[2.5 1.3 2.0 1.8 2.7 3.2 3.0 3.5 3.4];
nx=length(X);
ny=length(Y);
vx=var(X);
vy=var(Y);
#The null hypothesis is: H0: vx!=vy (the standard is meet)
#The alternative hypothesis is: H1: vx=vy (the standard is not meet)
#Both tail test for u when sigma is unknown
#oneminusalpha = input("Confidence level: (between 0 and 1)");
oneminusalpha = 0.95;
alpha = 1-oneminusalpha;
[H, PVAL, CI, ZVALUE] = vartest2(X,Y,'alpha',alpha,'tail','both');
if H==1
printf("The null hypothesis is rejected\n\n");
printf("The data suggests that the standard is not meet\n\n");
else
printf("The null hypothesis is not rejected\n\n");
printf("The data suggests that the standard is meet\n\n");
endif
#The null hypothesis is: H0: mx!=my (the standard is meet)
#The alternative hypothesis is: H1: mx=my (the standard is not meet)
#Both tail test for u when sigma is unknown
[H, PVAL, CI, ZVALUE] = ttest2(X,Y,'alpha',alpha,'tail','both');
if H==1
printf("The null hypothesis is rejected\n\n");
printf("The data suggests that the standard is not meet\n\n");
else
printf("The null hypothesis is not rejected\n\n");
printf("The data suggests that the standard is meet\n\n");
endif
Anul 2/Semestrul 1/Probabilitati si statistica/test_2.m
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X=[1001.7 975.0 978.3 988.3 978.7 988.9 1000.3 979.2 968.9 983.5 999.2 985.6];
n=length(X);
#oneminusalpha = input("Confidence level: (between 0 and 1)");
oneminusalpha = 0.95;
alpha = 1-oneminusalpha;
#The null hypothesis is: H0: u=995 (it goes togheter with u>995, the standard is meet)
#The alternative hypothesis is: H1: u<995 (the standard is not meet)
#Left tail test for u when sigma is unknown
printf("This is a left tail test for the mean when sigma is inknown\n\n");
n0=995;
[H, PVAL, CI, ZVALUE] = ttest(X,n0,'alpha',alpha,'tail','left');
if H==1
printf("The null hypothesis is rejected\n\n");
printf("The data suggests that the standard is not meet\n\n");
else
printf("The null hypothesis is not rejected\n\n");
printf("The data suggests that the standard is meet\n\n");
endif
oneminusalpha = 0.99;
alpha = 1-oneminusalpha;
v1=((n-1)*var(X))/(chi2inv(1-alpha/2, n-1));
v2=((n-1)*var(X))/(chi2inv(alpha/2,n-1));
s1=sqrt(v1);
s2=sqrt(v2);
printf("Confidence interval for the theoretical standard variance is (%4.3f, %4.3f)\n", s1,s2);