Covariance knowning conditional distribution of the two variablesFor an excercise at my university I need to find Var(X) where X = ( Y t , Z t ) t knowing that Z ∼ N ( 0 , I 2 ) and Y | Z ∼ N ( B Z , I 4 ) given that B ∈ M 4 , 2 is a matrix and the I 2 and I 4 are the identitis. I've managed to find that V a r ( Y ) = I 4 + B t B and I also already know Var(Z), I am stuck at the C o v ( Y , Z ) = E [ Y Z t ]. Does someone know how to compute it?

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Covariance knowning conditional distribution of the two variablesFor an excercise at my university I need to find Var(X) where   X  =  (      Y    t    ,      Z    t        )    t   knowing that   Z  ∼  N  (  0  ,      I    2    ) and   Y      |    Z  ∼  N  (  B  Z  ,      I    4    ) given that   B  ∈      M          4      ,      2       is a matrix and the       I    2   and       I    4   are the identitis. I&#039;ve managed to find that   V  a  r  (  Y  )  =      I    4    +      B    t    B and I also already know Var(Z), I am stuck at the   C  o  v  (  Y  ,  Z  )  =  E  [  Y      Z    t    ]. Does someone know how to compute it?

falwsay · Accepted Answer

Step 1Sanity check:       B    t    B is   2  ×  2, so       I    4    +      B    t    B does not make sense.The law of total variance implies  Var  (  Y  )  =  E  [  Var  (  Y  ∣  Z  )  ]  +  Var  (  E  [  Y  ∣  X  ]  )  =      I    4    +  Var  (  B  Z  )  =      I    4    +  B  E  [  Z      Z    t    ]      B    t    =      I    4    +  B      B    t    .Step 2Similarly, the law of total covariance implies                    Cov        (        Y        ,        Z        )        =        E        [        Cov        (        Y        ,        Z        ∣        Z        )        ]        +        Cov        (        E        [        Y        ∣        Z        ]        ,        E        [        Z        ∣        Z        ]        )        =        0        +        B        =        B        .            Alternatively, letting   Y  =  B  Z  +  U where   U  ∼  N  (  0  ,      I    4    ) is independent of Z,                    Cov        (        Y        ,        Z        )                            =        E        [        (        Y        −        E        [        Y        ]        )        (        Z        −        E        [        Z        ]                  )          t                ]                                          =        E        [        Y                  Z          t                ]                                          =        E        [        E        [        Y                  Z          t                ∣        Z        ]        ]                                          =        E        [        (        B        Z        +        U        )                  Z          t                ]                                          =        B        E        [        Z                  Z          t                ]        +        B        E        [        U                  Z          t                ]                                          =        B        .

Covariance knowning conditional distribution of the two variables. For an excercise at my university I need to find Var(X) where X=(Y^t, Z^t)^t knowing that Z sim N(0, I_2) and Y|Z sim N(BZ,I_4) given that B in M_{4,2} is a matrix and the I_2 and I_4 are the identitis.

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