Variance of a function with two random variables

Question

I have a function $f$ which depends on two random variables $a$ and $y$: $f(x,y)$.

I want to calculate the variance of $f$ w.r.t. $x$ and $y$, i.e. $\mathrm{Var}(f(x,y))$.

I am wondering if I can expand it similar to the definition of variance for a single random variable, i.e. if $\mathrm{Var}(f(x,y)) = E_{x,y} [f(x,y)^2] - [E_{x,y}(f(x,t))]^2$ valid for two random variables as well?

Answer 1

$\newcommand{\Var}{\operatorname{Var}}$ I'm not clear what you're using "$E_{x,y}$" to mean here - people use notation like that to mean a few different things in different contexts.

Let $\Var(.)$ mean variance and $E(.)$ mean expectation.

The formula $\Var(Z)=E(Z^2)-E(Z)^2$ applies to all random variables $Z$ whose variance exists.

Note that every random variable can be written as a function of two other random variables. For example, let $X=Z/3$ and $Y=2Z/3$, and let $f$ be the function given by $f(a,b)=a+b$. Then $Z=f(X,Y)$.