prob theory & stochastic18

Page 1

Exam zone


Rand

Example

Ou ω= ω= ω= ω= ω= ω= Figu Random of functions Consid of expr d functions the the di_eren mem dis thro

Sample x 1(t ) = x 2(t ) = 1 x 3(t ) = t x 4(t ) = t x5(t ) = t 2 x 6(t ) = t 2 mapping , 2, 3, 4, 5, 6. Note

x i (t ), =

X ( t, ) that The in T 1. F this examp

w ha

that the sampl If t = t o 2 T _ R , the functions at this instan denoted, X t o ( ω) c a

X t o ( ω) 2 { 1, 1, o , t o , 2o , t 2o } . The f X t o ( ω) ( x ; t o )

= +

1 { δ( x + 1) + δ( x 1) + δ( x t o ) } 6 1 _ δ( x + t o ) + δ( x t 2o ) + δ( x + t 2o ) . 6

Note ense

t . The

µx ( t ) = E { X t ( ω) } = The mean of the random v me In me Th σx2 ( t ) = E { X t2 ( ω) } = Note pling dep

1 A 6

1+ t

t + t2

t 2) =

X t ( ω), i.e., µx ( t ) constitutes the ense X ( t ) is

1 1 1 + 1 + t2 + t2 + t4 + t4_ = A 6 3

t . This non 1

t 2 + t 4).


pro Let t 1, 2, t1 < 2 _ , X 1 = X t 1 ( ω) an X 2 = X t 2 ( ω). Since the s pro f X 1 ,X 2 ( x 1 ,

2; t 1, 2)

= + +

1 { δ( x 1 6 1 { δ( x 1 6 1 _ δ( x 1 6

1,

This t the one X 1 is

f

of the underlying 2 is

X 1,

1) + δ( x 1 + 1,

2

2

+ 1)}

t 1,

2

t 2 ) + δ( x 1 + t 1 ,

2

+ t 2)}

t 21 ,

2

t 22 ) + δ( x 1 + t 21 ,

2

+ t 22 )

t 1 to

t 2 du

If

f X 1 (x1; t 1)

= +

1 { δ( x + 1) + δ( x 1) + δ( x t 1 ) } 6 1 _δ( x + t 1 ) + δ( x t 21 ) + δ( x + t 21 ) . 6

An mation X 1 and X 2 are not s ran f X 1 ,X 2 ( x 1 ,

X 2. F indep 2; t 1, 2)

b

the join P

of these

6 = f X 1 (x1 ; t 1 )f X 2 (x2 ; t 2 )

The R xx ( t 1 , 2 )

1 _A)A) 6

=

E { X 1 X 2_ } =

=

1 1 + t 1 t 2 + t 21 t 22 _ 3

1)( 1) + 2(t 1 )( t 2 ) + 2(t 21 t 22 )

Since function Cxx ( t 1 , 2 ) is the s as Rxx ( t 1 , 2 ). The temp tion that meas th normali stat correlati b v X 1 and X 2 is ρxx ( t 1 , 2 ) =

1 + t 1 t 2 + t 21 t 22 . p 1 + t 21 + t 41 p 1 + t 22 + t 42

Note pro sys

2

c t

fun random


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