Suppose we have a multivariate normal random variable X = [ X 1 , X 2 , X 3 , X 4 ] ⊤ . And here X 1 and X 4 are independent (not correlated). Also X 2 and X 4 are independent. But X 1 and X 2 are not independent. Assume that Y = [ Y 1 , Y 2 ] ⊤ is defined by Y 1 = X 1 + X 4 , Y 2 = X 2 − X 4 .If I know the covariance matrix of X, what would be the covariance matrix of Y?

Question

Suppose we have a multivariate normal random variable   X  =  [      X    1    ,      X    2    ,      X    3    ,      X    4        ]          ⊤       . And here       X    1   and       X    4   are independent (not correlated). Also       X    2   and       X    4   are independent. But       X    1   and       X    2   are not independent. Assume that   Y  =  [      Y    1    ,      Y    2        ]          ⊤       is defined by      Y    1    =      X    1    +      X    4    ,            Y    2    =      X    2    −      X    4    .If I know the covariance matrix of X, what would be the covariance matrix of Y?

Edward Chase · Accepted Answer

Step 1You can assume w.l.o.g. that   E  [  X  ]  =  0 . Then   E  [  Y  ]  =  0 (variances/covariances are not dependent on means).You need to compute   V  a  r  (      Y    1    )  ,  V  a  r  (      Y    2    )  ,  E  [      Y    1        Y    2    ] since the covariance matrix of Y is comprised of these three elements.Since       X    1    ,      X    4   are independent,   V  a  r  (      Y    1    )  =  V  a  r  (      X    1    )  +  V  a  r  (      X    4    ) which you should konw from the covariance matrix of X.Similarly for   V  a  r  (      Y    2    ) .Finally you can compute   E  [      Y    1        Y    2    ]  =  E  [      X    1        X    2    −      X    4    2    ]  =  C  o  v  (      X    1    ,      X    2    )  −  V  a  r  (      X    4    ) which you should know from the covariance matrix of X

Covariance matrix for multivariate normal random variable

Answered question

Answer & Explanation

New Questions in Inferential Statistics