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					            EE 505
Random Processes - Example Random Processes




                copyright Robert J. Marks II
Example RP’s
Example Random Processes
  Gaussian

 Recall Gaussian pdf
                            1                 
               1           ( x  m )T K 1 ( x  m )
fX (x)                  e 2
           2 
              n/2    1/ 2
                      |K|

Let Xk=X(tk) , 1  k  n. Then if, for all n, the
  corresponding pdf’s are Gaussian, then the
  RP is Gaussian.
The Gaussian RP is a useful model in signal
  processing.
                   copyright Robert J. Marks II
Flip Theorem
Let A take on values of +1 and -1 with equal
probability
Let X(t) have mean m(t) and
autocorrelation RX
Let Y(t)=AX(t)
Then Y(t) has mean zero and
autocorrelation RX
What about the autocovariances?
             copyright Robert J. Marks II
 Multiple RP’s
                     X(t) & Y(t)
   Independence
(X(t1), X(t2), …, X(tk ))
    is independent to
     (Y(1), Y( 2), …, Y( j ))

…for All choices of k and j and
all sample locations


                    copyright Robert J. Marks II
    Multiple RP’s
                   X(t) & Y(t)
   Cross Correlation
                RXY(t,  )=E[X(t)Y()]
   Cross-Covariance
        CXY(t,  )= RXY(t,  ) - E[X(t)] E[Y()]
   Orthogonal: RXY(t,  ) = 0
   Uncorrelated: CXY(t,  ) = 0
   Note: Independent Uncorrelated, but not
    the converse.
                 copyright Robert J. Marks II
  Example RP’s
Multiple Random Process
Examples
 Example

  X(t) = cos(t+), Y(t) = sin(t+),
  Both are zero mean.
  Cross Correlation=?
        p.338




                copyright Robert J. Marks II
    Example RP’s
Multiple Random Process Examples
   Signal + Noise
    X(t) = signal, N(t) = noise
    Y(t) = X(t) + N(t)
    If X & N are independent,RXY=?               p.338
    Note: also, var Y = var X + var N

                     var X
               SNR 
                     var N

                  copyright Robert J. Marks II
       Example RP’s
Multiple Random Process Examples (cont)
 Discrete time RP’s
     X[n]
     Mean
     Variance
     Autocorrelation
     Autocovariance
   Discrete time i.i.d. RP’s
      Bernoulli RP’s Binomial RP’s                   p.340
            Binary vs. Bipolar
      Random Walk       p.341-2




                        copyright Robert J. Marks II
Autocovariance of Sum Processes
                    n
              Sn   X [ k ]
                   k 1
X[k]’s are iid.

        E [ S n ]  nX

        var[ S n ]  n var( X )

Autocovariance=?


                   copyright Robert J. Marks II
Autocovariance of Sum Processes
                                        
   CS (n, k )  E ( S n  S n )(S k  S k )
               E ( S n  nX )(S k  kX )
                   n
                                 k
                                              
                                                
               E  ( X i  X ) ( X j  X )
                   i 1
                                j 1
                                              
                                                
When i=j, the answer is var(X). Otherwise, zero.
How many cases are there where i = j?
  min( n, k )  CS ( n , k )  min( n , k ) var( X )

                    copyright Robert J. Marks II
Autocovariance of Sum Processes
For Bernoulli sum process,
 var( X )  pq
 CS ( n , k )  min( n , k ) pq

For Bipolar case
  var( X )  4 pq
   CS ( n , k )  4 min( n , k ) pq


                     copyright Robert J. Marks II
Continuous Random Processes
Poisson Random Process
   Place n points randomly on line of length T
                            t

                        T
   Choose any subinterval of length t.
   The probability of finding k points on the
    subinterval is
                        n  k nk     t
    Pr[ k po int s ]    p q
                       k         ;p
                                     T
                 copyright Robert J. Marks II
Continuous Random Processes
Poisson Random Process (cont)
   The Poisson approximation: For k big and p small…

                n  k nk     np (np)
                                        k
Pr[k points]    p q
               k         e
                                   k!
                              (nt / T ) k
                  e  nt / T
                                  k!


                  copyright Robert J. Marks II
   Continuous Random Processes
  The Poisson Approximation…
      For n big and p small (implies k << n since p k/n<<1)
              n  k nk     np (np)
                                      k
              p q
             k         e
                                 k!
       Here’s why…
n       n!        n(n  1)(n  2)...(n  k  1) n k
 
 k  k!(n  k )!                               
                               k!                k!

        q n k  (1  p)nk  (1  p)n  (e p )n
                      copyright Robert J. Marks II
Continuous Random Processes
Poisson Random Process (cont)
                n  k nk          (np) k
Pr[k points]    p q
               k          e  np
                                    k!
                                                  k
                        nt / T (nt / T )
                  e
                                          k!
   Let n  such that =n/T = frequency of points
    remains constant.

                                       t (t )
                                                 k
     Pr[ k points on interval t ]  e
                                                      k!
                   copyright Robert J. Marks II
Continuous Random Processes
Poisson Random Process (cont)
                                                          t   ( t )   k
 Pr[ k points on interval t ]  e
                                                                   k!
   This is a Poisson process with parameter
                  occurrences per unit time
   Examples: Modeling
      Popcorn
      Rain (Both in space and time)
                                                     t
      Passing cars
      Shot noise
      Packet arrival times

                      copyright Robert J. Marks II
Continuous Random Processes
Poisson Counting Process
 X(t )




     Poisson Points
                                                 t   ( t )   k
                        Pr[ X ( t )  k ]  e
               copyright Robert J. Marks II
                                                          k!
 Continuous Random Processes
Recall for Poisson RV with parameter a
                             k
                  a   (a)
Pr[ X  k ]  e                           X  var( X )  a
                        k!
Poisson Counting Process Expected Value is
thus

          E [ X ( t )]  t

                   copyright Robert J. Marks II
Continuous Random Processes
The Poisson Counting Process is independent
increment process. Thus, for   t and j  i,

     Pr[ X (t )  i, X ( )  j ]
      Pr[X (t )  i, X ( )  X (t )  j  i ]
      Pr[X (t )  i ] Pr[ X ( )  X (t )  j  i ]

     
       t i et   (  t ) j i e ( t )
             i!                        ( j  i )!

                  copyright Robert J. Marks II
   Continuous Random Processes
                                                        t           
  Autocorrelation: If  > t
R X ( t , )  E X ( t ) X (  )
    
 E X ( t ) X (  )  X ( t )  X 2 ( t )             
 E X ( t ) X (  )  X ( t )  E X 2 ( t )            
 E X ( t )E X (  )  X ( t )  E X ( t )         2
                                                                
                     
 t  (   t )  t  t          2
                                           
  t  t
    2
                                  RX ( t , )  2t   min( t , )
                         copyright Robert J. Marks II
  Continuous Random Processes
 Autocovariance of a Poisson sum process


C X ( t , )  R X ( t , )  E  X ( t )E  X (  )
              t   min( t , )  t  
                 2

              min( t , )



                     copyright Robert J. Marks II
Continuous Random Processes
Other RP’s related to the Poisson process
   Random telegraph signal

       X(t )




               Poisson Points
                  copyright Robert J. Marks II
     Poisson Random Processes
    Random telegraph signal
                 2 |t |
E[ X (t )]  e
                   2|t  |
C X ( t , )  e

  PROOF…




                    copyright Robert J. Marks II
       Poisson Random Processes
      Random telegraph signal. For t>0,

E[ X (t )]  1 PrX (t )  1  (1)  Pr[X (t )  1]
 Prnumber of points on ( 0,t) is even
          Prnumber of points on ( 0,t) is odd




                      copyright Robert J. Marks II
      Poisson Random Processes
     Random telegraph signal. For t>0,

Prnumber of points on ( 0,t) is even
     t
          (t ) 2 (t ) 4      
 e 1                   ...
                                 
            2!      4!         
 e t cosht 




                     copyright Robert J. Marks II
       Poisson Random Processes
    Random telegraph signal. For t>0.
   Similarly…

Prnumber of points on ( 0,t) is odd
          
      t       (t )3 (t )5      
  e        t                ...
                  3!     5!        
                                   
           e  t sinh(t )



                      copyright Robert J. Marks II
     Poisson Random Processes
 Random telegraph signal. For t>0.
 Thus
E[ X (t )]  1 PrX (t )  1  (1)  Pr[ X (t )  1]
 Prnumber of points on ( 0 ,t) is even
          Prnumber of points on ( 0 ,t) is odd
 e  t cosh(t )  sinh(t )
 e  2t ; t  0
                                                      2 |t |
                    For all t…       E[ X (t )]  e
                     copyright Robert J. Marks II
     Poisson Random Processes
   Random telegraph signal. For t > ,
                             X()
                      1


                -1                   1     X(t)

                               -1

    Pr[ X (t ) X ( )  1]  PrX (t )  1, X ( )  1
          PrX (t )  1, X ( )  1
     PrX (t )  1 | X ( )  1Pr X ( )  1
          PrX (t )  1 | X ( )  1Pr X ( )  1
                     copyright Robert J. Marks II
    Poisson Random Processes
   Random telegraph signal. For t > ,

             PrX (t )  1 | X ( )  1
           PrX (t )  1 | X ( )  1
     Prnumber of points on ( , t ) is even
              e  ( t  ) cosh (t   ) 
Thus…
         PrX (t )  1, X ( )  1
          PrX (t )  1 | X ( )  1Pr X ( )  1
          cosh (t   ) e  (t  ) cosh( )e 

                   copyright Robert J. Marks II
         Poisson Random Processes
        Random telegraph signal. For t > ,
  PrX (t )  1, X ( )  1
   PrX (t )  1 | X ( )  1Pr X ( )  1
                                                                X()
   cosh (t   ) e t cosh( )                        1


    And…
                                                       -1              1   X(t)
PrX (t )  1, X ( )  1
                                                                 -1
 PrX (t )  1 | X ( )  1Pr X ( )  1
 cosh (t   ) e t sinh( )

                        copyright Robert J. Marks II
        Poisson Random Processes
       Random telegraph signal. For t > .
Onward…

PrX (t )  1 | X ( )  1
 Pr X (t )  1 | X ( )  1
 Prnumber of points on ( , t ) is odd
 e  ( t  ) sinh  (t   ) 


                           copyright Robert J. Marks II
        Poisson Random Processes
       Random telegraph signal. For t > .
PrX (t )  1, X ( )  1
 PrX (t )  1 | X ( )  1Pr X ( )  1                  X()
                                                           1
 sinh  (t   ) e t cosh( )
And…
                                                      -1              1   X(t)
PrX (t )  1, X ( )  1
                                                                -1
 PrX (t )  1 | X ( )  1Pr X ( )  1
 sinh  (t   ) e t sinh( )

                       copyright Robert J. Marks II
        Poisson Random Processes
       Random telegraph signal. For t > .
PrX (t )  1, X ( )  1
 PrX (t )  1 | X ( )  1Pr X ( )  1                  X()
                                                           1
 sinh  (t   ) e t cosh( )
And…
                                                      -1              1   X(t)
PrX (t )  1, X ( )  1
                                                                -1
 PrX (t )  1 | X ( )  1Pr X ( )  1
 sinh  (t   ) e t sinh( )

                       copyright Robert J. Marks II
Poisson Random Processes
   Random telegraph signal. For t > .                                    X()
                                                                       1


                                                                  -1              1   X(t)

     RX (t , )  EX (t ) X ( )                                          -1

      1 PrX (t ) X ( )  1  1 PrX (t ) X ( )  1
       
      cosh (t   ) e t cosh( )  sinh (t   ) e t sinh( )           
      sinh (t   ) e   t
                                   cosh( )  cosh (t   ) e t       sinh( )

    In general… X (t , )  RX (t , )  X (t ) X ( )  e2 |t  |
               C
                                   copyright Robert J. Marks II
Continuous Random Processes
Other RP’s related to the Poisson process
 Poisson point process, Z(t)
Let X(t) be a Poisson sum process. Then
                   d
          Z ( t )  X ( t )    ( t  Sn )
                   dt         n
         Z( t )


pp.352

                  Poisson Points
                      copyright Robert J. Marks II
Continuous Random Processes
Other RP’s related to the Poisson process
 Shot Noise, V(t)
Z(t)                     V(t)
             h(t)
                               V ( t )   h( t  S n )
                                                    n
    V( t )


pp.352
              Poisson Points
                     copyright Robert J. Marks II
    Continuous Random Processes
Wiener Process
   Assume bipolar Bernoulli sum process with jump
    bilateral height h and time interval 
   E[X(t)]=0; Var X(n) = 4npqh 2 = nh 2
   Take limit as h 0 and   0 keeping  = h 2 / 
    constant and t = n .
   Then Var X(t)   t
   By the central limit theorem, X(t) is Gaussian with zero
    mean and Var X(t) =  t
   We could use any zero mean process to generate the
    Wiener process.
p.355
                      copyright Robert J. Marks II
Continuous Random Processes
Wiener Processes:  =1




               copyright Robert J. Marks II
  Continuous Random Processes
  Wiener processes in finance
S= Price of a Security.  = inflationary force.
If there is no risk…interest earned is proportional to investment.
                                       dS
                dS( t )  S ( t )dt      S
                           t          dt
Solution is S( t )  S0e
  With “volatility” , we have the most commonly used model in
  finance for a security:
        dS( t )  S( t )dt  S( t )dV ( t )
   V(t) is a Wiener process.

                      copyright Robert J. Marks II

				
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posted:9/24/2012
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