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Covariance Matrix of a Vector Random Variable w/ Components Related

  1. Jul 2, 2013 #1
    1. The problem statement, all variables and given/known data
    Given X=ZU+Y
    where
    (i) U,X,Y, and Z are random variables
    (ii) U~N(0,1)
    (iii) U is independent of Z and Y
    (iv) f(z) = [itex]\frac{3}{4}[/itex] z2 if 1 [itex]\leq z[/itex] [itex]\leq 2 [/itex], f(z)=0 otherwise
    (v) fY|Z=z(y) = ze-zy (i.e. Y depends conditionally on Z)
    (vi) define vector W=[X Y Z]T

    Q: What is the covariance matrix KWW?


    2. Relevant equations
    KWW = E[(W-E[W])(W-E[W])T]
    KWW = [itex]\begin{pmatrix}K_{XX} & K_{XY} & K_{XZ} \\
    K_{YX} & K_{YY} & K_{YZ} \\
    K_{ZX} & K_{ZY} & K_{ZZ}
    \end{pmatrix}
    [/itex]

    [itex]V[X] = E[(X-E[X])^2] = E[X^2] - E^2[X] [/itex]
    [itex]V[X] = E_Z[V[X|Z=z]] + V_Z[E[X|Z=z]] [/itex] (Total Variance Theorem)

    3. The attempt at a solution
    I can calculate using [itex]K_{ZZ} = E[Z^2] - E^2[Z][/itex], and the definition of expectation.
    However, I'm having difficulty calculating the other terms.

    For example, if I use the Total Variance Theorem for KXX I get the following:
    (using Total Variance Theorem)
    [itex]V[X] = E_Z[V[X|Z=z]] + V_Z[E[X|Z=z]] [/itex]
    (using X=ZU+Y)
    [itex]V[X] = E_Z[V[ZU+Y|Z=z]] + V_Z[E[ZU+Y|Z=z]] [/itex]
    (using Z=z)
    [itex]V[X] = E_Z[V[zU+Y|Z=z]] + V_Z[E[zU+Y|Z=z]] [/itex]
    (using V[aX+bY|Z=z]=a2V[X|Z=z]+b2V[Y|Z=z]+2abCov[X,Y|Z=z])
    [itex]V[X] = E_Z[z^2V[U|Z=z] + V[Y|Z=z] + 2zCov[U,Y|Z=z]] +V_Z[E[zU+Y|Z=z]] [/itex]
    (using E[aX+bY|Z=z]=aE[X|Z=z] +bE[Y|Z=z])
    [itex]V[X] = E_Z[z^2V[U|Z=z] + V[Y|Z=z] + 2zCov[U,Y|Z=z]] +V_Z[zE[U|Z=z] + zE[Y|Z=z]] [/itex]
    (using U,Z are independent, for U|Z=z terms drop the Z=z)
    [itex]V[X] = E_Z[z^2V + V[Y|Z=z] + 2zCov[U,Y|Z=z]] +V_Z[zE + zE[Y|Z=z]] [/itex]
    (using E= 0 and V=1 which is given by U~N(0,1) )
    [itex]V[X] = E_Z[z^2 + V[Y|Z=z] + 2zCov[U,Y|Z=z]] +V_Z[0 + zE[Y|Z=z]] [/itex]

    Now, I can find [itex]V[Y|Z=z][/itex] and [itex]E[Y|Z=z]] [/itex], but I'm not sure how to find Cov[U,Y|Z=z]].

    From my understanding U independent of Y means Cov[U,Y]=0, but not necessarily that Cov[U,Y|Z=z] = 0. Let me know if this last statement is incorrect.

    I'm I approaching this correctly?
    Is there an easier way?
     
  2. jcsd
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