Why does this expectation equality hold?

Question

$$E[Y] = \int_0^\infty P\{Y>y\} dy$$

It's said that $\int_0^\infty P\{Y>y\} dy = \int_0^\infty (\int_0^x dy) f_Y(x) dx$, but I don't know why it is $0$ and $x$ in the parentheses which I thought should be $y$ and $\infty$.

Any suggestions would be highly appreciated. Thanks.

Answer 1

if $Y\geq 0$

$$\int_0^\infty P\{Y>y\} dy = \int_0^\infty (\int_y^\infty f_Y(x) dx) dy$$

$$=\iint_{0<y<\infty \, y<x<\infty} f_Y(x) dx dy$$

$$=\iint_{0<x<\infty \, 0<y<x} f_Y(x) dx dy$$ $$= \int_0^\infty (\int_0^x dy) f_Y(x) dx$$

$$= \int_0^\infty (x) f_Y(x) dx=\int_0^\infty x f_Y(x) dx=E(Y)$$

Area $$0<y<\infty \hspace{1cm} y<x<\infty$$ equals $$0<x<\infty \hspace{1cm} 0<y<x$$