Generating Random Numbers

Mean, Variance, Standard Deviation

1

n

ii

x

nm ==

å• Mean:

– mean(x)

• Variance:– mean((x-mean(x).*(x-mean(x)))

• Standard Deviation– std(x)

2

2 1

( )n

ii

x

n

ms =

-=å

2

1

( )n

ii

x

n

ms =

-=å

Correlation Coefficient

• Correlation coefficient

function r = corco(x,y)

mx = mean(x);

my = mean(y);

covxy = mean((x-mx).*(y-my));

r = covxy/(std(x)*std(y));

• Or use Matlab function– corrcoef

XY

X Y

sr

s s=

Random Numbers

• rand(M,N): MxN matrix of uniformly distributed random numbers on (0,1)

• randn(M,N) MxN matrix of normally distributed random numbers (μ=0, σ2=1)

• normrnd(m,s,M,N) MxN matrix of normally distributed random numbers (μ=m, σ=s)

x=randn(1,10000);mean((x<=1))ans = 0.8449

(1) 0.8413F =

Correlation Coefficient

(0,1)X N

(0,1)N N

Y aX b

Y aX b N

1

0.9186

x=normrnd(0,1,1,100);

y=2*x+1;

r=corrcoef(x.',y.')

plot(x,y,'+')

n=normrnd(0,1,1,100);

y=2*x+1+n;

r=corrcoef(x.',y.')

plot(x,y,'+')

Joint Gaussian

2 21 2 1 2

2 2 21 21 2

( ) ( ) 2 ( )( )1

2(1 ), 2

1 2

1( , )

2 (1 )

x m y m x m y m

X Yf x y e

0

Joint Gaussian

0.5

Joint Gaussian

0.9

Joint Gaussian

0.9

Generating Random Numbers

1. Generate

2. Return

Uniform

1( )f x a x b

b a

ìïï= £ £íï -ïî

0

( )

1

x a

x aF x a x b

b ab x

ì <ïïïï -ïï= £ £íï -ïïï <ïïî

(0,1)U U:

( )X a b a U= + -

Uniform

function genuni(N,a,b)

u=rand(1,N);

x=a+(b-a).*u;

minx=min(x);

maxx=max(x);

NumBins=51;

h=hist(x,NumBins);

for k=1:NumBins,

bincenters(k)=minx+((maxx-minx)/NumBins)*(k-1/2);

end

h=h/sum(h);

bar(bincenters,h)

10 15a b= =

Or use Matlab function unifrnd(a,b,M,N)

Generating Random Numbers

1. Generate

2. Return

{( ) 0xf x e xll -= ³1 0

( )0 0

xe xF x

x

l-ìï - ³ï=íï <ïî

(0,1)U U:1

ln(1 )X Ul

=- -

Exponential

Exponential

function genexp(N,lambda)

u=rand(1,N);

x=-1/lambda.*log(1-u);

5l =

Or use Matlab function exprnd(lambda,M,N)

Generating Random Numbers

2

2

( )

21

( )2

x

f x e xm

s

ps

--

= - ¥ < <¥

Normal

2

2

( )

21

( )2

x x

F x e xm

s

ps

--

- ¥

= - ¥ < <¥ò

2

222

( ) , 0xx

f x e xs

s

-= ³

2

221 0( )0 0

x

e xF xx

s-ìïïï - ³=íïï <ïî

1. Generate

2. Set

3. Generate

4. Return

1 (0,1)U U:2

1

12 ln

1Z

Us

æ ö÷ç ÷= ç ÷ç ÷ç -è ø2 (0,1)U U:

1 2 2 2cos(2 ) sin(2 )X Z U X Z Um p m p= + = +

Rayleigh

Normal

210 4m s= =function gennormal(N,mu,sigma)

for i=1:N

u=rand; z=sigma*(sqrt(2*log(1/(1-u))));

u=rand;

x1(i)=mu+z*cos(2*pi*u);

x2(i)=mu+z*sin(2*pi*u);

end

Or use Matlab function normrnd(mu,sigma,M,N)

Generating Random Numbers

Binomial

1. Generate IID Bernoulli(p) random numbers

2. Return1 2, ,..., nY Y Y

1 2 ... nX Y Y Y= + + +

( ) (1 ) 0,1,...,k n knp k p p k n

k-

æö÷ç ÷= - =ç ÷ç ÷çè ø 0

0 0

( ) (1 ) 0

1

ki n i

i

k

nF k p p k n

i

k n

-

=

ì <ïïïï æöïï ÷ç ÷= - £ £í ç ÷ç ÷çï è øïïï >ïïî

å

Binomial

20 0.5n p= =function genbino(N,n,p)

for i=1:N,

u=rand(1,n);

y=(u<=p);

x(i)=sum(y);

end

Or use Matlab function binornd(n,p,M,N)

Generating Random Numbers

Geometric

1( ) (1 ) 1,2,...kp k p p k-= - =1 (1 ) 1

( )0 1

kp kF k

k

ìï - - ³ï=íï <ïî

1. Generate

2. Return

(0,1)U U:ln(1 )

ln(1 )

UX

p

é ù-ê ú=ê ú-ê ú

Geometric

0.5p=function gengeo(N,p)

u=rand(1,N);

x=ceil(log(1-u)/log(1-p));

Or use Matlab function geornd(p,M,N)

Generating Random Numbers

Poisson

( ) , 0,1,...!

k

p k e kk

ll -= =

0

0 0

( )0

!

ik

i

k

F ke k

ill -

=

ì <ïïïï=íï ³ïïïîå

1. Set

2. Generate and replace by

3. If accept else increase by one and return to step 2

0, 1k P= =

1 (0,1)kU U+ : P 1kP U +×

P e l-< X k= k

Poisson

5l =function genpois(N,lambda)

for i=1:N,

k=0;p=1;

u=rand;

p=p*u;

while p>=exp(-lambda)

u=rand;

p=p*u;

k=k+1;

end

x(i)=k;

end

Or use Matlab function poissrnd(lambda,M,N)