### 02_distributions

```Some Useful Distributions
Binomial Distribution
ænö
k
n- k
÷
p(k ) = çç ÷
p
(1
p
)
,
÷
çèk ø
÷
m = np
k = 0,1,..., n
Bernoulli
1720
s 2 = np(1- p)
k=0:20;
y=binopdf(k,20,0.5);
stem(k,y)
n = 20
p = 0.5
k=0:20;
y=binocdf(k,20,0.5);
stairs(k,y)
grid on
Binomial Distribution
function y=mybinomial(n,p)
for k=0:n
y(k+1)=factorial(n)/(factorial(k)*factorial(n-k))*p^k*(1-p)^(n-k)
end
k=0:20;
y=mybinomial(20,0.5);
stem(k,y)
n = 20
p = 0.5
k=0:20;
y=binopdf(k,20,0.1);
stem(k,y)
n = 20
p = 0.1
Geometric Distribution
p(k ) = (1- p)k- 1 p,
1
m=
p
1- p
s =
p2
k = 1, 2,...
Warning: Matlab assumes
p(k ) = (1- p)k p,
2
k=0:20;
y=geopdf(k,0.5);
stem(k,y)
p = 0.5
k=0:20;
y=geocdf(k,0.5);
stairs(k,y)
axis([0 20 0 1])
k = 0,1, 2,...
Geometric Distribution
function y=mygeometric(n,p)
for k=1:n
y(k)=(1-p)^(k-1)*p;
end
k=1:20;
y=mygeometric(20,0.5);
stem(k,y)
p = 0.5
k=1:20;
y=mygeometric(20,0.1);
stem(k,y)
p = 0.1
Poisson Distribution
l k -l
p(k ) =
e ,
k!
m= l
k = 0,1,...
Poisson 1837
s2= l
k=0:20;
y=poisspdf(k,5);
stem(k,y)
l =5
k=0:20;
y=poisscdf(k,5);
stem(k,y)
grid on
Poisson Distribution
function y=mypoisson(n,lambda)
for k=0:n
y(k+1)=lambda^k/factorial(k)*exp(-lambda);
end
k=0:10;
y=mypoisson(10,0.1);
stem(k,y)
axis([-1 10 0 1])
l = 0.1
k=0:10;
y=mypoisson(10,2);
stem(k,y)
axis([-1 10 0 1])
l = 2
Uniform Distribution
f ( x) =
1
,
b- a
a+ b
m=
2
a£ x£ b
(b - a)2
s =
12
2
x=0:0.1:8;
y=unifpdf(x,2,6);
plot(x,y)
axis([0 8 0 0.5])
x=0:0.1:8;
y=unifcdf(x,2,6);
plot(x,y)
axis([0 8 0 2])
Normal Distribution
f ( x) =
1
e
2ps
-
( x- m)2
2s 2
,
- ¥ < x< ¥
Gauss 1820
N (m, s 2 )
Warning: Matlab uses
x=0:0.1:20;
y=normpdf(x,10,2);
plot(x,y)
N (10, 4)
N (m, s )
x=0:0.1:20;
y=normcdf(x,10,2);
plot(x,y)
Normal Distribution
function y=mynormal(x,mu,sigma2)
y=1/sqrt(2*pi*sigma2)*exp(-(x-mu).^2/(2*sigma2));
x=-6:0.1:6;
y1=mynormal(x,0,1);
y2=mynormal(x,0,4);
plot(x,y1,x,y2,'r');
legend('N(0,1)','N(0,4)')
Exponential Distribution
f ( x) = l e- l x ,
1
m=
l
s2=
0£ x< ¥
1
l
x=0:0.1:5;
y=exppdf(x,1/2);
plot(x,y)
l = 2
2
Warning: Matlab assumes
1 - lc
f ( x) = e ,
l
0£ x< ¥
x=0:0.1:5;
y=expcdf(x,1/2);
plot(x,y)
Exponential Distribution
function y=myexp(x,lambda)
y=lambda*exp(-lambda*x);
x=0:0.1:10;
y1=myexp(x,2);
y2=myexp(x,0.5);
plot(x,y1,x,y2,'r')
legend('lampda=2','lambda=0.5')
Rayleigh Distribution
f ( x) =
x
e
2
s
m= s
p
2
-
x2
2s 2
,
x³ 0
æ pö 2
s 2 = çç2 - ÷
s
÷
÷
çè
2ø
x=0:0.1:10;
y1=raylpdf(x,1);
y2=raylpdf(x,2);
plot(x,y1,x,y2,'r')
legend('sigma=1','sigma=2')
Poisson Approximation to Binomial
n? 1
p= 1
np = l
k
ænö
l
k
n
k
-l
çç ÷
÷
p
(1
p
)
;
e
÷
çèk ø
÷
k!
n=100;
p=0.1;
lambda=10;
k=0:n;
y1=mybinomial(n,p);
y2=mypoisson(n,lambda);
stem(k,y1)
hold on
stem(k,y2,’r’)
Normal Approximation to Binomial
DeMoivre – Laplace Theorem 1730
If X is a binomial RV
normal RV
Z=
ænö
k
n- k
÷
P[ X = k ] = çç ÷
p
(1
p
)
;
÷
÷
çèk ø
X - np
np(1- p)
is approximately a standard
1
e
2p np(1- p)
-
( k - np )2
2 np (1- p )
æ k - np ö
æ k - np ö
æn÷
ö k
÷
÷
ç
çç 1
n- k
2
ç
÷
÷
ç
P[k1 £ X £ k2 ] = å ç ÷
p
(1
p
)
;
F
F
÷
÷
÷
ç
ç
÷
÷
÷
ç
k
ç
ç
è
ø
np
(1
p
)
np
(1
p
)
k = k1
è
ø
è
ø
k2
A better approximation
æk + 0.5 - np ÷
ö
æk - 0.5 - np ÷
ö
ç
ç
2
1
÷
÷
P[k1 £ X £ k2 ] ; F çç
- F çç
÷
÷
÷
çè np(1- p) ø
èç np(1- p) ÷
ø
Normal Approximation to Binomial
function normbin(n,p)
clf
n = 10
y1=mybinomial(n,p);
p = 0.5
k=0:n;
bar(k,y1,1,'w')
hold on
x=0:0.1:n;
y2=mynormal(x,n*p,n*p*(1-p));
plot(x,y2,'r')
n = 30
p = 0.5
Central Limit Theorem
function k=clt(n) % Central Limit Theorem for sum of dies
m=(1+6)/2;
% mean (a+b)/2
s=sqrt(35/12); % standart deviation sqrt(((b-a+1)^2-1)/12)
for i=1:n
x(i,:)=floor(6*rand(1,10000)+1);
end
for i=1:length(x(1,:))
% sum of n dies
y(i)=sum(x(:,i));
z(i)=(sum(x(:,i))-n*m)/(s*sqrt(n));
end
subplot(2,1,1)
hist(y,100)
title('unormalized')
subplot(2,1,2)
hist(z,100)
title('normalized')
Central Limit Theorem
m=
a+ b
= 3.5
2
(b - a + 1) 2 - 1
s =
= 2.92
12
2
n= 1
n
unormalized =
å
Xi
i= 1
n
å
normalized =
X i - nm
i= 1
s n
n= 2
Central Limit Theorem
n= 5
n = 10
```