How can I develop this?

Question

$$\operatorname{var}(X - Y)^2 $$ How can I develop this? I think I should arrive at a covariancy, but I can't think of any property.

Answer 1

Assuming you mean var($ (X-Y)^2 $), that is var ($X^2 - 2XY + Y^2$). Repeated use of V(A+B) = V(A) + 2 Cov(A,B) + V(B) as well as V(aX) = $a^2$V(X) gives me V($X^2$) -4 Cov($X^2, XY$) + 2Cov($X^2, Y^2$) + 4V(XY) - 4Cov($XY,Y^2$) + V($Y^2$). There is no simple relationship between V(X) and V($X^2$) etc. For example if X were normal, $X^2$ would be a $\chi ^2$ random variable.