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Sampling and histograms. x= randn (1,5000); xcenters = linspace (-3.5,3.5,8) xcenters = -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5. x= randn (1,5000); xcenters = linspace (-3.5,3.5,8) ;. hist ( x,xcenters ). hist ( x,xcenters ). More boxes. x= randn (1,5000);

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sampling and histograms
Sampling and histograms

x=randn(1,5000); xcenters=linspace(-3.5,3.5,8)

xcenters = -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5

x=randn(1,5000); xcenters=linspace(-3.5,3.5,8)

;

hist(x,xcenters)

  • hist(x,xcenters)
more boxes
More boxes
  • x=randn(1,5000);
  • hist(x,20)
  • x=randn(1,5000):
  • Hist(x,20)

)

cdf with only 500 samples
CDF with only 500 samples

z=randn(1,500);[f,x]=ecdf(z); plot(x,f); hold on

X=[-4:0.1:4];p=normcdf(X,0,1);plot(X,p,'r');

xlabel('x'); ylabel('normal CDF')

Repeat z=randn(1,500);[f,x]=ecdf(z); plot(x,f); three more times

probability plot
Probability plot
  • z=randn(1,500); probplot(z)
  • Repeat (hold off)
lognormal distribution
Lognormal distribution
  • Mode (highest point) =
  • Median (50% of samples)
  • Mean =
  • Figure for =0.
lognormal probability plot
Lognormal probability plot

z=lognrnd(0,1,1,500);probplot('lognormal',z)

probplot(z)

the normalizing effect of averaging
The normalizing effect of averaging

z=lognrnd(0,1,100,500);

zmean=mean(z);

probplot(zmean)

mean(zmean')

=1.6395

%Exact mean=exp(0.5)= 1.648

std(zmean')=

0.2088

%Original standard deviation=sqrt(exp(1)-1)*exp(1))=2.1612

fitting a distribution
Fitting a distribution

x=randn(20,1)+3; [ecd,xe,elo,eup]=ecdf(x);

pd=fitdist(x,'normal')

pd = NormalDistribution

Normal distribution

mu = 2.89147 [2.36524, 3.41771]

sigma = 1.1244 [0.855093, 1.64226]

xd=linspace(1,8,1000);

cdfnorm=normcdf(xd, 2.89147, 1.1244);

plot(xe,ecd,'LineWidth',2); hold on;

plot(xd,cdfnorm,'r','LineWidth',2); xlabel('x');ylabel('CDF')

plot(xe,elo,'LineWidth',1); plot(xe,eup,'LineWidth',1)

fit lognormal instead
Fit lognormal instead

pd=fitdist(x,'lognormal')

pd = LognormalDistribution

Lognormal distribution

mu = 0.973822 [0.759473, 1.18817]

sigma = 0.457998 [0.348303, 0.668939]

cdflogn=logncdf(xd,0.973822,0.45799); hold on; plot(xd,cdflogn,'g','LineWidth',2)

with more points it is clearer
With more points it is clearer

Same as before, but with 200 points

x=randn(200,1)+3; [ecd,xe,elo,eup]=ecdf(x);

pd=fitdist(x,'normal')

Normal distribution

mu = 3.00311 [2.87373, 3.13248]

sigma = 0.927844 [0.844952, 1.02891]

pd=fitdist(x,'lognormal')

Lognormal distribution

mu = 1.04529 [0.996772, 1.09382]

sigma = 0.347988 [0.3169, 0.385893]