Download Simple Stochastic Models 2

Simple stochastic models 2
Continuous random variable X
X – can take values: - < x < +
Cumulative probability distribution function:
PX(x) = P(X  x)
Probability density function:
P ( x  X  x  x )
p( x)  lim
x  0
x
Normal distribution
X ~ normal(,), E(X)= , V(X)=  2
pdf:
1
( x   ) 2 /( 2 2 )
p ( x) 
e
 2
1
cpdf: P( x) 
 2
x
e

 ( u   ) 2 /( 2 2 )
du
=0, =1
0.4
0.3
pdf
0.2
0.1
0
-4
-3
-2
-1
0
1
2
3
4
-3
-2
-1
0
1
2
3
4
1
0.8
cpdf
0.6
0.4
0.2
0
-4
x
Central limit theorem
Y = X1 + X2 + … +Xn
Xi - independent, zero mean, equal variance V
If n is large then:
Y
nV
~ normal(0,1)
Example – students body heights
Histogram of students’
body heights versus
normal probability
density function
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
160
165
170
175
180
185
190
195
200
205
Example – children birth weights
9
Histogram of children
birth weights versus
normal probability
density function
x 10-4
8
7
6
5
4
3
2
1
0
0
1000
2000
3000
4000
5000
6000
Binomial becomes normal
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
15
o – binomial(0.5,50)
20
25
30
35
40
45
50
normal(25,3.5355)
How do we fit normal
distribution to data ?
Data: X1, X2, …, Xn
1 n
mean   X i
n i 1
1 n
2
std 
(
X

mean
)

i
n  1 i 1
• How do we estimate parameters of
distributions using data ?
• How do we verify that data follow a given
distribution ?
Characteristic function
X – with pdf p(x)
characteristic function:

 ( )   e
j x
p( x)dx

 ( )  E e
j X

j  1
Properties
 ( 0)  1
 ' (0)  jE( X )
 ' ' (0)   E ( X 2 )
Properties
 X ( )  E e
j X

Y=aX+b, a,b - constants
Y ( )  Ee
j  ( aX b )
 e
j b
 X (a )
Characteristic function of normal
distribution
X ~ normal(,),
1
( x   ) 2 /( 2 2 )
p ( x) 
e
 2
 ( )  e
 2 / 2 2  j / 
Continuity theorem for
characteristic functions
Two dimensional distributions
X, Y
Probability density function: p(x,y)
b1 b2
P(a1  X  b1 , a2  Y  b2 )    p( x, y )dxdy
a1 a2
Cumulative pdf:
x y
P ( X  x, Y  y ) 
  p(u, v)dudv
  
Independent random variables
X, Y independent
pXY(x,y)=pX(x) pY(y)
Convolution integral Z=X+Y

pZ ( z ) 
p


X
( z  y ) pY ( y )dy   p X ( x) pY ( z  x)dx

p Z = pX * p Y
Convolution and characteristic
functions
X ~  X ( )
Y ~ Y ( )
Z=X+Y
Z ( )   X ( )Y ( )
Use of characteristic functions to
prove Central Limit Theorem
Y = X1 + X2 + … +Xn
Y
nV
 ( )  E e j  X
so: Y ( )   ( )
n
i

i=1,2…,n
 

)
and  Y ( )   (
nV 

nV
n

V
1
2
(
)  1
  o 
2nV
nV
n
 
1 2

n log  (
)   
2
nV 
