Calculating $\mathbb{E}X^2$

Question

When we could use the following equation: $$\mathbb{E}X^2=\int_0^\infty 2t \mathbb{P}(X>t)$$ I mean how is it possible to change $X^2$ to $2t$?

Answer 1

Assuming $X$ is nonnegative, the trick is to (1) put $x^2=\int_0^x2t\,dt$ in the formula for $\mathbb E(X^2)$, obtaining a double integral, and then (2) interchange the order of integration. Writing $f(x)$ for the density of $X$, the derivation goes: $$ \begin{align} \mathbb E(X^2)&=\int_{x=0}^\infty x^2f(x)\,dx\\ &\stackrel{(1)}=\int_{x=0}^\infty\left(\int_{t=0}^x2t\,dt\right)f(x)\,dx\\ &=\int_{x=0}^\infty\left(\int_{t=0}^x2tf(x)\,dt\right)\,dx\\ &\stackrel{(2)}=\int_{t=0}^\infty\left(\int_{x=t}^\infty2tf(x)\,dx\right)\,dt\\ &=\int_{t=0}^\infty 2t\left(\int_{x=t}^\infty f(x)\,dx\right)\,dt\\ &=\int_{t=0}^\infty 2t\,\mathbb P(X>t)\,dt\\ \end{align} $$