Tail integral formula for moments ($\mathbb{E}|X{|}^q=q{\int }_0^{\infty }{t}^{q-1}\mathbb{P}(|X|>t)dt$)

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 $\mathbb{E}{X}^2=2{\int }_0^{\infty }t{e}^{-\lambda t}dt=2/{\lambda }^2$.

For a nonnegative random variable, $Y={\int }_0^{\infty }{1}_{\{Y>t\}}dt$ pointwise, including when $Y=\infty$. Tonelli's theorem permits taking expectation under the integral and proves the formula in the extended nonnegative reals. This relates finiteness of a Brownian exit time expectation to integrability of its survival probability.

If $\mathbb{P}(|Z|\ge t)\le A{e}^{-b{t}^2}$ for $A\ge 1$ and $b>0$, then $\mathbb{E}{Z}^2\le (1+\log A)/b$. Indeed the tail integral formula for moments gives $\mathbb{E}{Z}^2\le {\int }_0^{\infty }\min (1,A{e}^{-bu})du$; split at $u=(\log A)/b$. Clipping at one improves the constant compared with integrating the exponential bound alone.