Binomial Distribution
k=0:20;
y=binopdf(k,20,0.5);
stem(k,y)
( ) (1 ) , 0,1,...,k n knp k p p k n
k-
æö÷ç ÷= - =ç ÷ç ÷çè ø
2 (1 )np np pm s= = -
20 0.5n p= =
Bernoulli 1720
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)
k=0:20;
y=binopdf(k,20,0.1);
stem(k,y)
20 0.5n p= = 20 0.1n p= =
Geometric Distribution
k=0:20;
y=geopdf(k,0.5);
stem(k,y)
1( ) (1 ) , 1,2,...kp k p p k-= - =
22
1 1 p
p pm s
-= =
0.5p=
k=0:20;
y=geocdf(k,0.5);
stairs(k,y)
axis([0 20 0 1])
( ) (1 ) , 0,1,2,...kp k p p k= - =
Warning: Matlab assumes
Geometric Distribution
k=1:20;
y=mygeometric(20,0.5);
stem(k,y)
k=1:20;
y=mygeometric(20,0.1);
stem(k,y)
function y=mygeometric(n,p)
for k=1:n
y(k)=(1-p)^(k-1)*p;
end
0.5p= 0.1p=
Poisson Distribution
k=0:20;
y=poisspdf(k,5);
stem(k,y)
( ) , 0,1,...!
k
p k e kk
ll -= =
2m l s l= =
k=0:20;
y=poisscdf(k,5);
stem(k,y)
grid on
5l =
Poisson 1837
Poisson Distribution
k=0:10;
y=mypoisson(10,0.1);
stem(k,y)
axis([-1 10 0 1])
function y=mypoisson(n,lambda)
for k=0:n
y(k+1)=lambda^k/factorial(k)*exp(-lambda);
end
0.1l =
k=0:10;
y=mypoisson(10,2);
stem(k,y)
axis([-1 10 0 1])
2l =
Uniform Distribution
1( ) ,f x a x b
b a= £ £
-2
2 ( )
2 12
a b b am s
+ -= =
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
Gauss 18202
2
( )
21
( ) ,2
x
f x e xm
s
ps
--
= - ¥ < <¥
x=0:0.1:20;
y=normpdf(x,10,2);
plot(x,y)
2( , )N ms
x=0:0.1:20;
y=normcdf(x,10,2);
plot(x,y)
(10,4)N
Warning: Matlab uses ( , )N ms
Normal Distribution
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)')
function y=mynormal(x,mu,sigma2)
y=1/sqrt(2*pi*sigma2)*exp(-(x-mu).^2/(2*sigma2));
Exponential Distribution
( ) , 0xf x e xll -= £ <¥
22
1 1m s
l l= =
2l =
x=0:0.1:5;
y=exppdf(x,1/2);
plot(x,y)
Warning: Matlab assumes1
( ) , 0f x e xcl
l
-= £ <¥
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
2
222
( ) , 0xx
f x e xs
s
-= ³
2 222 2
p pm s s s
æ ö÷ç= = - ÷ç ÷÷çè ø
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=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’)
1 1n p np l=? =
(1 )!
kk n knp p e
k kll- -
æö÷ç ÷ -ç ÷ç ÷çè ø;
Normal Approximation to Binomial
2( )
2 (1 )1[ ] (1 )
2 (1 )
k npk n k np pn
P X k p p ek np pp
--
- -æö÷ç ÷= = -ç ÷ç ÷çè ø -
;
DeMoivre – Laplace Theorem 1730
(1 )
X npZ
np p
-=
-If X is a binomial RV is approximately a
standard normal RV
A better approximation
2
1
2 11 2[ ] (1 )
(1 ) (1 )
kk n k
k k
n k np k npP k X k p p
k np p np p-
=
æ ö æ öæö - -÷ ÷ç ç÷ç ÷ ÷ç ç÷£ £ = - F - Fç ÷ ÷÷ ç çç ÷ ÷÷ç ÷ ÷ç çç çè ø - -è ø è øå ;
2 11 2
0.5 0.5[ ]
(1 ) (1 )
k np k npP k X k
np p np p
æ ö æ ö+ - - -÷ ÷ç ç÷ ÷ç ç£ £ F - F÷ ÷ç ç÷ ÷÷ ÷ç çç ç- -è ø è ø;
Normal Approximation to Binomial
10
0.5
n
p
=
=
30
0.5
n
p
=
=
function normbin(n,p)
clf
y1=mybinomial(n,p);
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')
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
1n=
2n=
3.52
a bm
+= =
22 ( 1) 1
2.9212
b as
- + -= =
1
n
ii
unormalized X=
=å
1
n
ii
X nnormalized
n
m
s=
-=å