Extremes of Random Numbers of Random Variables: A Survey

Abstract

Let X ₁ , X ₂ , ⋯, be independent, identically distributed random variables with distribution function F. Let N be a positive integer-valued random variable independent of the X _i ’s. Let ψ(u), 0 ≤ u ≤ 1, be the probability-generating function of N. It is easy to verify that

$$ G(x) \equiv \psi (F(x)),\quad - \infty < x < \infty , $$

(9.1)

is the distribution function of X _(N:N) ≡ max{X ₁, X ₂, …, X _N }, and that

$$ H(x) \equiv 1 - \psi (1 - F(x)),\quad - \infty < x < \infty , $$

(9.2)

is the distribution function of X _(1:N) ≡ min{X ₁, X ₂, …, X _N }. If for a distribution function F we denote $\bar{F} = 1 - F$, then (9.2) can also be written as

$$ \bar{H}(x) = \psi \left( {\bar{F}(x)} \right),\quad - \infty < x < \infty . $$

(9.3)