ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4689 on D149: Lebesgue norm

Probabilistic Cavalieri principle

Formulation 0

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 $P$

Let $p \in (0, \infty)$ be a D993: Real number.

Then \begin{equation} \mathbb{E} X^p = \int^{\infty}_0 \mathbb{P}(X > t) p t^{t - 1} \, d t \end{equation}

Formulation 1

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 $P$

Let $p \in (0, \infty)$ be a D993: Real number.

Then \begin{equation} \mathbb{E} X^p = \int^{\infty}_0 \mathbb{P}(X > t) \, d t^p \end{equation}

Proofs

Proof 0

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 $P$

Let $p \in (0, \infty)$ be a D993: Real number.

This result is a particular case of R112: Cavalieri principle. $\square$