WOLFRAM|DEMONSTRATIONS PROJECT

Distributions of Discrete Order Statistics

​
distribution
uniform
sample without replacement
binomial
geometric
Poisson
number of observations
3
i (
th
i
order statistics)
1
6 (uniform, binomial sample without replacement)
6
0.5 (binomial, geometric)
0.5
λ (Poisson)
3.
​
Let
X
1
, …,
X
n
be a random sample from a discrete distribution. Reorder the sample in increasing order; denote the corresponding variables by
Y
1
, …,
Y
n
. Thus, for example,
Y
1
is the smallest of the
X
variables,
Y
2
the second smallest, and
Y
n
the largest. The variable
Y
i
is called the
th
i
order statistic. This Demonstration shows the distributions of the order statistics (the red curves) when the sample is from a uniform, binomial, geometric, or Poisson distribution or when the sample is made without replacement from a finite set (the underlying distribution is shown in blue).