A Parameterizing conditional expectations (Gaussian case)

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1. Mar 2, 2016

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]=xßZX;Y +yßZY;X
where ßZX;Y notes the regression coefficient of Z on X conditional on Y (and ßZY;Xis analogously defined).

Is the following derivation correct?

E[Z| X>x, Y<y] = E [ E[Z | X,Y] | X>x, Y<y]
ZX;Y E[X | X>x, Y<y]+ßZY;X E[Y |X>x, Y<y]

2. Mar 2, 2016

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.

3. Mar 10, 2016

estebanox

Great! Thanks a lot for spelling it out