understanding the limits in calculation of expectation

Question

My question is from the book Bertsekas, "Introduction to probability".

Let's say X is continuous first

I believe I should maniputate the given expression to look the like known definiton of Expecation.

$ E[X] = \int_{-\infty}^\infty x \times f_X(x) dx $

equivalently,

$ \int_{0}^\infty P(X > x)dx = \int_{0}^\infty \Bigl(\int_{x}^\infty f_X(x) dx\Bigr) dx $

The solution is given below.

Why is a new variable y introduced? and why did the limits change from $(x,\infty)$ to $(0,y)$

Thanks.

Answer 1

The solution uses the definition of the tail probability for a nonnegative random variable $P(x>x)=\int_x^\infty f(\tau) d\tau$. The next step is switching the order of integration (Fubini's theorem).