I'm trying to figure out something but I can't seem to get it.
My professor wrote in the notes relative to E[Y] = $\int_{0}^{\infty}$P(Y>y) if Y is a takes non-negative values:
Let the indicator function be := $1_B(x)$ = $1$ if $x\in B$ and 0therwise for B a borel set.
Then, let $Y:\Omega \rightarrow \mathbb{R}$ the random variable with density function $1_B(y)$
It implies that
P{Y>y} = $\int_{0}^{\infty} 1_{Y>y}(y) dy = y$
I'm trying to find sense to that. Because from what I understand, it's the probability that $y>y$ which kind of never happens. But if I change say $x$ instead of y for one of them. Then when $x$ reaches $y$, we get
$\int_{x}^{\infty} 1dy$ = y????
Anyway, if anyone could help me make some sense to that, I would appreciate it very much.