Parameterizing conditional expectations (Gaussian case)

[estebanox]

Consider three jointly normally distributed random variables X,Y and Z.

I know that in the Gaussian case E[Z | X,=x Y=y]=xß_ZX;Y +yß_ZY;X
where ß_ZX;Y notes the regression coefficient of Z on X conditional on Y (and ß_ZY;Xis analogously defined).

Is the following derivation correct?

E[Z| X>x, Y<y] = E [ E[Z | X,Y] | X>x, Y<y]
=ß_ZX;Y E[X | X>x, Y<y]+ß_ZY;X E[Y |X>x, Y<y]

 

[andrewkirk]

Yes it is correct.
The first equality is an application of the Tower Law.
The second equality actually comprises two steps:
1. substitute your formula for E[Z | X,Y] from above for the inner expectation
2. Use the linearity of the expectation operator and the fact that the betas are constants to get the result.

 

[estebanox]

Yes it is correct.
The first equality is an application of the Tower Law.
The second equality actually comprises two steps:
1. substitute your formula for E[Z | X,Y] from above for the inner expectation
2. Use the linearity of the expectation operator and the fact that the betas are constants to get the result.

Great! Thanks a lot for spelling it out