I have two question about random variable that make me confused.
1.A max function means that the maximum element from a set.
but what if the set contain some random variable,such as $Max(X,Y)$. where $X$,$Y$ are random variable. how do I interpret this function.
2.I define a function $G(k,X)$=$Max(0,k-X)$ where $k$ is a constant and $X$ is a random variable.
Is that two statement true?
1.for $k \in \Bbb R$ ,function $G(k,．)$ is convex
2.for any random variable $X$ ,function $G(．,X)$ is convex
$\max(X,Y)$ is a random variable; when the value of $X$ is $x$ and the value of $Y$ is $y$, the value of $\max(X,Y)$ is $\max(x,y)$.
$G(k, \cdot)$ is a function taking random variables to random variables. We have to be careful about what it means for such a thing to be to be convex: presumably, it means that for all random variables $X$ and $Y$ and all $t \in [0,1]$, $$G(k, tX + (1-t)Y) \le t G(k,X) + (1-t) G(k,Y) \ \text{almost surely}$$
And indeed this is true since for all $x$ and $y$, $G(k, tx + (1-t)y) \le t G(k,x) + (1-t) G(k,y)$.
Similarly, $G(\cdot, X)$ is a function taking real numbers to random variables; its convexity means
$$G(tk_1 + (1-t) k_2, X) \le tG(k_1, X) + (1-t)G(k_2, X)$$ which is true because the corresponding statement with $X$ replaced by any value $x$ is true.