 Tail integral formula for moments

For $q>0$ and a real random variable $X$, the tail integral formula for moments isvalid even if both sides are infinite. Pointwise, $|X{|}^q={\int }_0^{|X|}q{t}^{q-1}dt$; Tonelli theorem interchanges this nonnegative integral with expectation. Replacing $>$ by $\ge$ does not change the integral, since their difference is confined to the at most countably many atoms of the distribution. For an exponential random variable of rate $\lambda$, the formula at $q=2$ gives $\mathbb{E}{X}^2=2{\int }_0^{\infty }t{e}^{-\lambda t}dt=2/{\lambda }^2$.