Distribution of the Sample Range of Discrete Random Variables

distribution

uniform

binomial

geometric

Poisson

number of variables

2

m (uniform, binomial)

6

p (binomial, geometric)

0.166667

λ (Poisson)

1

Let

X

1

, …,

X

n

be a random sample from a probability distribution. Let

Y

1

and

Y

n

be the smallest and the largest value in the sample. The sample range

W

is

W=

Y

n

-

Y

1

, that is, the difference between the largest and the smallest value. The sample range is a crude measure of the variation in the sample. The Demonstration shows the distribution of the sample range, when the sample is drawn from some well-known discrete distributions.