How to treat generally the function of random variable(s)?

Question

I did not really understand the functions of $X+Y$, $X^2$, etc. For example, $F(X^2)$, $f(X+Y)$, etc. How should I treat them? Can you explain the logic behind it?

Answer 1

You should treat them as random variables. What's a random variable $X$? It's a map from the sample space to $\mathbb{R}$, $X:\Omega \to \mathbb{R}$. What's $Y = F(X)$? It's a map $Y: \Omega \to^{X} \mathbb{R} \to^{F} \mathbb{R}$, so it's a random variable.