Suppose Z ( t ) = Σ k = 1 n X e j ( 𝜔 0 t + 𝚽 k ) , t ∈ R where 𝜔 0 is a constant, n is a fixed positive integer, X 1 , . . . , X n , 　 𝚽 1 , . . . , 𝚽 n are mutually independent random variables, and E X k = 0 , D X k = σ k 2 , 𝚽 , U [ 0 , 2 π ] , k = 1 , 2 , . . . , n . Find the mean function and correlation function of { Z ( t ) , t ∈ R } .I have tried to solve it.For mean function, m Z ( s ) = E { Z s } = E { X s } + i E { Y t } = E { Σ k = 1 s X e j ( 𝜔 0 t + 𝚽 k ) }For correlation function, R Z ( s , u ) = E { Z s , Z u } = E { Y ( s ) Y ( u ) } = E { Σ k = 1 s X e j ( 𝜔 0 t + 𝚽 k ) Σ k = 1 u X e j ( 𝜔 0 t + 𝚽 k ) } = E { Σ k = 1 s Σ k = 1 u X e j ( 𝜔 0 t + 𝚽 k ) X e j ( 𝜔 0 t + 𝚽 k ) }I am stuck here. How to move from here ahead?

Question

Suppose   Z  (  t  )  =            Σ              k      =      1              n        X      e          j      (                        𝜔                0            t      +                        𝚽                k            )      ,   t  ∈  R where             𝜔        0   is a constant, n is a fixed positive integer,       X    1    ,  .  .  .  ,      X    n    ,      　              𝚽        1    ,  .  .  .  ,            𝚽        n   are mutually independent random variables, and   E      X    k    =  0  ,  D      X    k    =            σ              k              2        ,      𝚽   ,   U  [  0  ,  2      π    ]  ,  k  =  1  ,  2  ,  .  .  .  ,  n . Find the mean function and correlation function of   {  Z  (  t  )  ,     t  ∈  R  } .I have tried to solve it.For mean function,      m    Z    (  s  )  =  E  {      Z    s    }  =  E  {      X    s    }  +  i  E  {      Y    t    }  =  E  {            Σ              k      =      1              s        X      e          j      (                        𝜔                0            t      +                        𝚽                k            )        }For correlation function,      R    Z    (  s  ,  u  )  =  E  {      Z    s    ,      Z    u    }  =  E  {  Y  (  s  )  Y  (  u  )  }  =  E  {            Σ              k      =      1              s        X      e          j      (                        𝜔                0            t      +                        𝚽                k            )                  Σ              k      =      1              u        X      e          j      (                        𝜔                0            t      +                        𝚽                k            )        }  =  E  {            Σ              k      =      1              s                  Σ              k      =      1              u        X      e          j      (                        𝜔                0            t      +                        𝚽                k            )        X      e          j      (                        𝜔                0            t      +                        𝚽                k            )        }I am stuck here. How to move from here ahead?

suponeriq · Accepted Answer

Step 1Before finding the mean and variance function, one has to fix the notation of   Z  (  t  ), which should be  Z  (  t  )  =      ∑          k      =      1              n            X    k        e          j      (                        𝜔                0            t      +              Φ        k            )        . .In order to compute the mean of   Z  (  t  ), it suffices to compute the expectation of       X    k        e          j      (                        𝜔                0            t      +              Φ        k            )       . To do so, use the independence between       X    k   and       Φ    k   and the fact that       X    k   is centered.For the correlation function, the most technical part is to compute   Cov  ⁡      (    Z    (    s    )    ,    Z    (    t    )    )   which actually reduces to      ∑          k      =      1              n            ∑          ℓ      =      1        n        E        [          X      k              X      ℓ              e              j        (                  ω          0                s        +                  Φ          k                )                    e              j        (                  ω          0                t        +                  Φ          ℓ                )              ]  (when you have to sums, it is always preferable to sum over different indices). If   k  ≠  ℓ , the corresponding term vanishes.

Suppose Z(t)=sum_{k=1}^{n}Xe^{j(omega_{0}t+Phi_{k})}, t in R where omega_{0} is a constant...

Answered question

Answer & Explanation

New Questions in Inferential Statistics