Utilisateur:Mathieumerca/One-sided Vysochanskii-Petunin inequality

In probability theory, the Vysochanskii–Petunin inequality gives a lower bound for the probability that a random variable with finite variance lies within a certain number of standard deviations of the variable's mean, or equivalently an upper bound for the probability that it lies further away. The sole restrictions on the distribution are that it be unimodal and have finite variance. The theorem applies even to heavily skewed distributions and puts bounds on how much of the data is, or is not, "in the middle." However, for many applications, for instance in the financial area, one-sided inequalities are most relevant in the sense of "how bad can losses get." Thus, for a unimodal random variable X, the one-sided Vysochanskii-Petunin inequality holds as follows:

$\mathbb {P} (X-E[X]\geq r)\leq \%{\begin{cases}{\dfrac {4}{9}}{\dfrac {V(X)}{r^{2}+V(X)}}&{\mbox{for }}r^{2}\geq {\dfrac {5}{3}}V(X),\\{\dfrac {4}{3}}{\dfrac {V(X)}{r^{2}+V(X)}}-{\dfrac {1}{3}}&{\mbox{otherwise.}}\end{cases}}$ Changez ce texte pour votre brouillon.