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