# How to think about a function of random variables?

Consider the random variables $X, Y$ defined on the probability space $(S, \mathcal{A}, \mu)$ taking value respectively in the Borel spaces $(\mathcal{X}, \mathcal{B}_{\mathcal{X}})$, $(\mathcal{Y}, \mathcal{B}_{\mathcal{Y}})$.

Consider a measurable function $f:\mathcal{X}\times \mathcal{Y}\rightarrow \mathbb{R}$.

Consider $f(X,Y)$.

How should I think about $f(X,Y)$?

(1) $f(X,Y)$ can be thought of as a random variable $W:S\rightarrow \mathbb{R}$ such that $W(s)=f(X(s), Y(s))$.

(2) $f(X,Y)$ can be thought of as a deterministic function $f:\mathcal{X}\times \mathcal{Y}\rightarrow \mathbb{R}$ assigning $f(x,y)$ to each $(x,y)\in \mathcal{X}\times \mathcal{Y}$.

(3) Can $f(X,Y)$ be thought as a random function, i.e. as a random variable taking value in a function space?

• Hints: (2) is a description of $f$. Can you find any function space in this context? – drhab Apr 26 '17 at 14:37
• So is mine. Btw, only (1) is correct. – drhab Apr 26 '17 at 14:43
• Because it is not a description of $f(X,Y)$. As said it is a description of $f$. – drhab Apr 26 '17 at 14:45
• Is $f(X,Y)$ the same as $f$...no. That should address (2) and (3). Think about the objects you are modeling, not how they are generated. – user408433 Apr 26 '17 at 14:46
• Yes, that makes sense. For example, if$G(s) = X(s)y$, then indeed $G(S)$ will generate linear functions of $y$ with random slopes. – user408433 Apr 26 '17 at 15:08