Let $X \in \text{Random} [0, \infty]$ be a D5101: Random unsigned basic number such that

(i)	$F : \mathbb{R} \to [0, 1]$ is a D205: Probability distribution function for $X$

Result R3956: Expectation of random unsigned basic number by integrating tail probabilities shows that \begin{equation} \mathbb{E} X = \int^{\infty}_0 \mathbb{P}(X > t) \, d t \end{equation} By definition, $F(t) : = \mathbb{P}(X \leq t)$. Applying result R3558: Probability of level sets for random basic number, we may thus conclude that \begin{equation} \mathbb{E} X = \int^{\infty}_0 \mathbb{P}(X > t) \, d t = \int^{\infty}_0 (1 - \mathbb{P}(X \leq t)) \, d t = \int^{\infty}_0 (1 - F(t)) \, d t \end{equation}