Teorema Bapat-Beg

Sea ${\displaystyle X_{1},X_{2},\ldots ,X_{n}}$ variables aleatorias reales independientes con funciones de distribución acumulativa respectivamente ${\displaystyle F_{1}(x),F_{2}(x),\ldots ,F_{n}(x)}$. Escribe ${\displaystyle X_{(1)},X_{(2)},\ldots ,X_{(n)}}$ para las estadísticas de pedido. Entonces la distribución de probabilidad de ${\displaystyle n_{1},n_{2},\ldots ,n_{k}}$ estadísticas de pedidos (con ${\displaystyle n_{1}<n_{2}<\cdots <n_{k}}$ and ${\displaystyle x_{1}<x_{2}<\cdots <x_{k}}$) es

${\displaystyle {\begin{aligned}F_{X_{(n_{1})},\ldots ,X_{(n_{k})}}(x_{1},\ldots ,x_{k})&=\Pr(X_{(n_{1})}\leq x_{1}\land X_{(n_{2})}\leq x_{2}\land \cdots \land X_{(n_{k})}\leq x_{k})\\&=\sum _{i_{k}=n_{k}}^{n}\cdots \sum _{i_{2}=n_{2}}^{i_{3}}\sum _{i_{1}=n_{1}}^{i_{2}}{\frac {P_{i_{1},\ldots ,i_{k}}(x_{1},\ldots ,x_{k})}{i_{1}!(i_{2}-i_{1})!\cdots (n-i_{k})!}},\end{aligned}}}$

donde:

${\displaystyle {\begin{aligned}&P_{i_{1},\ldots ,i_{k}}(x_{1},\ldots ,x_{k})={}\\[5pt]&\operatorname {per} {\begin{bmatrix}F_{1}(x_{1})\cdots F_{1}(x_{1})&F_{1}(x_{2})-F_{1}(x_{1})\cdots F_{1}(x_{2})-F_{1}(x_{1})&\cdots &1-F_{1}(x_{k})\cdots 1-F_{1}(x_{k})\\F_{2}(x_{1})\cdots F_{2}(x_{1})&F_{2}(x_{2})-F_{2}(x_{1})\cdots F_{2}(x_{2})-F_{2}(x_{1})&\cdots &1-F_{2}(x_{k})\cdots 1-F_{1}(x_{k})\\\vdots &\vdots &&\vdots \\\underbrace {F_{n}(x_{1})\cdots F_{n}(x_{1})} _{i_{1}}&\underbrace {F_{n}(x_{2})-F_{n}(x_{1})\cdots F_{n}(x_{2})-F_{n}(x_{1})} _{i_{2}-i_{1}}&\cdots &\underbrace {1-F_{n}(x_{k})\cdots 1-F_{n}(x_{k})} _{n-i_{k}}\end{bmatrix}}\end{aligned}}}$

es el permanente de la matriz por bloques dada. (Las cifras debajo de los tirantes muestran el número de columnas.)[1]