Order statistic

For two independent rate-$\lambda >0$ exponential distributions, set $U=\min (X,Y)$ and $V=\max (X,Y)$. Summing the two ordering branches gives joint probability density $2{\lambda }^2{e}^{-\lambda (2u+d)}$ for $(U,d=V-U)$ with $u,d>0$. It factors into independent exponential probability density functions. Thus $V$ is distributed as $X+Y/2$, with expected value $3/(2\lambda )$ and variance $5/(4{\lambda }^2)$. This spacing result is the order-statistic manifestation of the memorylessness of the exponential distribution.

Independence factors the events that both variables exceed a threshold or both lie below it. This gives the two cumulative distribution functions above, including for distributions with atoms. The minimum formula is most naturally expressed through survival probabilities, while the maximum formula uses the CDFs directly.

Descending order statistics list a sample from greatest to least. If $V_{(1)}\le \cdots \le V_{(n)}$ uses the usual increasing convention, then $V_{[k]}=V_{(n-k+1)}$. This notation distinguishes the next-highest value used by revenue equivalence from the second-smallest sample value.

For independent and identically distributed random variables with distribution function $F$, the sample maximum $M_n={\max }_{1\le i\le n}{X}_i$ has distribution function $F(x{)}^n$.

For observations $X_1,\dots ,X_n$, the nearest neighbour distance of order $k$ at $x$ is the $k$th order statistic of the distances $|X_i-x|$. An atomless distance distribution function ensures ties occur with probability zero.

For independent and identically distributed random variables with continuous distance distribution function $p_x$, the probability integral transform makes the probability content of the order-$k$ nearest ball a uniform order statistic, with the Beta distribution $\mathrm{Beta}(k,n+1-k)$.

For a strictly positive probability density function with Lipschitz bound $L$, $p_x(r)={\int }_{x-r}^{x+r}f(y)dy$ satisfies $|p_x(r)-2rf(x)|\le L{r}^2$. If $0<s<1$ and $s\le f(x{)}^2/L$, monotonicity gives $p_x^{-1}(s)\le s/f(x)$ and hence $|2f(x)p_x^{-1}(s)-s|\le L{s}^2/f(x{)}^2$. The Lipschitz density height bound prevents extending this local argument to every $s\in (0,1)$ by assuming $f(x)\ge \sqrt{L}$ for a strictly positive density.

For independent uniform random variables $U_1,\dots ,U_n$ on $[0,1]$, the $j$th order statistic has distribution $\mathrm{Beta}(j,n-j+1)$ and mean $j/(n+1)$.

Independent and identically distributed random variables with a continuous probability distribution realize each of their possible strict orderings with equal probability.