Starting with the code in Histograms.m generate the following plot. Use 100,000 random normal numbers with a standard deviation of 5 and a mean of three. Plot red vertical lines for sigma=-3:3, and label the z scores as shown. Call this mfile DistributionHomework_LastName.m.

Histograms.m

clc, clear, close all, %close all hidden

%% dist 1 Uniform distribution

Rnd1=rand(10000,1);

figure(100)

hist(Rnd1,100)

a=axis;

axis(a+[-.2 .21 -10 0]);

grid on

Men1=mean(Rnd1)

Std1=std(Rnd1)

Max1=max(Rnd1)

Min1=min(Rnd1)

Max1-1

% note that the numbers are evenly distributed between 0 and 1

% while the max displays as one you can see that it is less than 1

% Note that the mean is aprox 0.5 and the std is 0.289

%% dist 2 normal distribution

Rnd2=randn(10000,1);

figure(200)

hist(Rnd2,100)

grid on

Men2=mean(Rnd2)

Std2=std(Rnd2)

Max2=max(Rnd2)

Min2=min(Rnd2)

Max2-1

% mean is aprox 0, std is 1 min is -3.36, max is 4.28

% what kind of distrigution is this?

%% dist 3 binomial dist

Rnd3=binornd(100,0.2,100000,1);

figure(300)

hist(Rnd3,100)

grid on

Men3=mean(Rnd3)

Std3=std(Rnd3)

Max3=max(Rnd3)

Min3=min(Rnd3)

% mean is 20 = .2*100, std is 4

% what is the big difference? Look at the values in Rnd3

% what kind of distrigution is this?

%% Pdf normal

Rnd4=randn(100000,1);

[Cnt,Bin]=hist(Rnd4,1000);

Pdf=Cnt/sum(Cnt);

BinWid=mean(diff(Bin));

figure(400)

bar(Bin,Pdf/BinWid)

grid on

%% Cdf and Erf

Cdf1=cumsum(Pdf);

Erf=1/2+1/2*erf(Bin/sqrt(2));

Erfc=1/2*erfc(-Bin/sqrt(2));

NrmCdf=normcdf(Bin); % qfunc not available in 2014

figure(410)

plot(Bin, Cdf1,'b')

hold on

plot(Bin,Erf,'r')

plot(Bin,NrmCdf,'m')

plot(Bin,Cdf1-Erf+0.5,'g')

plot(Bin,(Cdf1-Erf).*Cdf1*100+0.5,'c')

hold off

grid on

%% Z, z=(x-xBar)/xSig

xSig=3;

xBar=10;

Rnd5=randn(100000,1)*xSig+xBar;

[Cnt5,Bin5]=hist(Rnd5,1000);

Pdf5=Cnt5/sum(Cnt5);

BinWid5=mean(diff(Bin5));

figure(500)

bar(Bin5,Pdf5/BinWid5);

grid on

a=axis; %a=[xmin xmax ymin ymax];

% add mean and +/- 1 sig

hold on

plot([xBar xBar], a(3:4), 'r')

plot([xBar xBar]-xSig, a(3:4), 'g')

plot([xBar xBar]+xSig, a(3:4), 'g')

plot([xBar xBar]-xSig*2, a(3:4), 'g')

plot([xBar xBar]+xSig*2, a(3:4), 'g')

plot([xBar xBar]-xSig*3, a(3:4), 'g')

plot([xBar xBar]+xSig*3, a(3:4), 'g')

hold off

title('Sigma=3, Xbar=10');

%% find the probability that x<7

% first use pdf

fndx=Bin5<7;

p7a=sum(Pdf5(fndx));

% now use erf

% z=(x-xBar)/xSig

x7=7;

z7=(x7-xBar)/xSig;

p7b=1/2+1/2*erf(z7/sqrt(2));

disp([p7a,p7b, p7a-p7b])

%% replot with shaded area

figure(510)

bar(Bin5(~fndx),Pdf5(~fndx)/BinWid5,'c');

grid on

hold on

bar(Bin5(fndx),Pdf5(fndx)/BinWid5,'g');

plot([xBar xBar], a(3:4), 'r')

plot([xBar xBar]-xSig, a(3:4), 'g')

plot([xBar xBar]+xSig, a(3:4), 'g')

plot([xBar xBar]-xSig*2, a(3:4), 'g')

plot([xBar xBar]+xSig*2, a(3:4), 'g')

plot([xBar xBar]-xSig*3, a(3:4), 'g')

plot([xBar xBar]+xSig*3, a(3:4), 'g')

hold off

%% compute probability vs sigma

for kSig=1:6

Prb(kSig,1)=normcdf(kSig)-normcdf(-kSig);

%Prb(kSig,1)=normcdf(-kSig);

end

%Ap(Prb)