Claim: If (X, Y) ⊥ Z, then X ⊥ Z and Y ⊥ Z. True or False? If false, what about if (X,Y) are jointly Gaussian

Question

General Claim: If (X, Y) ⊥ Z, then Y ⊥ Z and X ⊥ Z. True or False?

If false, what about if (X,Y) are jointly Gaussian?

Specific problem I am dealing with: Consider a case of simple linear regression, where $$y_i = B_0 + B_1x_i + \sigma z_i \text{ ~ }_{i.i.d.} N(B_0 + B_1x_i, \sigma^2),i=1,…n$$

There is a result that $$(B_0,B_1)\perp \sum_i (y_i – (B_0 + B_1x_i))^2$$

I want to know if $$B_i\perp \sum_i (y_i – (B_0 + B_1x_i))^2$$

Answer 1

The general statement is correct.

Proof: $$ P(X=x,Z=z) = \int_Y P(X=x,Y=y,Z=z)dy = $$

$$\int_Y P(X=x,Y=y)P(Z=z)dy=P(X=x)P(Z=z) $$