ThmDex – An index of mathematical definitions, results, and conjectures.
Expectation with dual probability distribution function

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 $X$
Then $$\mathbb{E} (X) = \int^{\infty}_0 (1 - F (t)) \, d t$$

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $X : \Omega \to [0, \infty]$ is a D5101: Random unsigned basic number on $P$ (ii) $F : \mathbb{R} \to [0, 1]$ is a D205: Probability distribution function for $X$
Then $$\mathbb{E} (X) = \int^{\infty}_0 (1 - F (t)) \, d t$$
Proofs
Proof 0
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 $$\mathbb{E} X = \int^{\infty}_0 \mathbb{P}(X > t) \, d t$$ 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 $$\mathbb{E} X = \int^{\infty}_0 \mathbb{P}(X > t) \, d t = \int^{\infty}_0 (1 - \mathbb{P}(X \leq t)) \, d t = \int^{\infty}_0 (1 - F(t)) \, d t$$