Given that I am dealing with a discrete distribution, what exactly does the following notation mean:
$\mathbb{E}[X\mathbb{1}_{\{ X \geq x_q\}}]$ where $\mathbb{1}$ is an indicator function. How do I actually compute something like that.
Where $x_q = \text{sup} \{ x: P(X \geq x) \leq 1-q\}$. I actually have difficulty understanding this too. $q$ can take the value of say $90\%$ then we would have that $x_{90\%} = \text{sup} \{ x: P(X \geq x) \leq 0.1 \}$. I have previously encountered the following: $\text{VaR}_q=\text{inf}_x (P(X>x) \leq 1-q)$. And to me it is like the two are the same, but I know they are not..
EDIT: I think I got it (about $x_q$), but in such a case, this would not make sense $\mathbb{E}[X\mathbb{1}_{\{ X \geq x_q\}}]$, as we would be asking what is the expected value of the largest value in out discrete distribution? Which makes me think that my interpretation is wrong. Also, if $q=90$ then supremum is the same for every $q$ larger than $90$.. which also makes me think that my interpretation is incorrect