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).