Category:Lévy's Inversion Formula
Jump to navigation
Jump to search
This category contains pages concerning Lévy's Inversion Formula:
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $ F_X$ be the distribution function of $X$.
Let $\phi : \R \to \C$ be the characteristic function of $X$.
Then for all $a < b$ such that:
- $\map \Pr {X \in \set {a,b} } = 0$
we have:
| \(\ds \map {F_X} b - \map {F_X} a\) | \(=\) | \(\ds \map \Pr {a < X \le b}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim _{T \to \infty} \frac 1 {2 \pi} \int^T _{-T} \dfrac {e^{-ita} - e^{-itb} }{it} \map \phi t \rd t\) |
Source of Name
This entry was named for Paul Pierre Lévy.
Pages in category "Lévy's Inversion Formula"
The following 3 pages are in this category, out of 3 total.