If we know that c o r r ( X , Y ) = ρ and Y , Z are independent and also X , Y , Z ∼ N ( 0 , 1 ) then can we deduce anything about c o r r ( X , Z ) ?

Question

If we know that   c  o  r  r  (  X  ,  Y  )  =  ρ and   Y  ,  Z are independent and also   X  ,  Y  ,  Z  ∼  N  (  0  ,  1  ) then can we deduce anything about   c  o  r  r  (  X  ,  Z  ) ?

i4epdp · Accepted Answer

Step 1Clearly   corr  (  Y  ,  Z  )  =  0 . If   corr  (  X  ,  Y  )  =      ρ          X      Y                     and   corr  (  X  ,  Y  )  =      ρ          X      Y                      then the covaraince matrix is      [                            1                                      ρ                          X              Y                                                                                                  ρ                          X              Z                                                                                                              ρ                          X              Y                                                                                      1                          0                                                  ρ                          X              Z                                                                                      0                          1                      ]  A necessary condition for this to be positive semidefinite is that its determinant is non-negative, i.e.  1  −      ρ          X      Y        2    −      ρ          X      Z        2    ≥  0which implies  −      1    −          ρ              X        Y            2        ≤      ρ          X      Z                      ≤      1    −          ρ              X        Y            2      In general, that is not sufficient, but it is here since the other principal minors have determinants of 1 or   1  −      ρ          X      Y        2   or   1  −      ρ          X      Z        2   , which are all non-negative.As examples of all such values in this intervals, suppose we have another standard normal random variable W independent of Y and Z so we can construct   X  =      ρ          X      Y                      Y  +      ρ          X      Z                      Z  +      1    −          ρ              X        Y            2        −          ρ              X        Z            2        W . This will give the desired correlation matrix for   X  ,  Y  ,  Z , at least so long as       1    −          ρ              X        Y            2        −          ρ              X        Z            2       is real.

Correlation of IID normal RV's

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