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# Discrete Uniform Distribution

Authored by: K. Krishnamoorthy

# Handbook of Statistical Distributions with Applications

Print publication date:  October  2015
Online publication date:  January  2016

Print ISBN: 9781498741491
eBook ISBN: 9781498741507

10.1201/b19191-5

#### Abstract

The probability mass function of a discrete uniform random variable X is given by

#### 3.1  Description

The probability mass function of a discrete uniform random variable X is given by

$P ( X = k ) = 1 N , k = 1 , … , N .$

The cumulative distribution function is given by

$P ( X ≤ k ) = k N , K = 1 , … N .$

This distribution is used to model experimental outcomes that are “equally likely.” The mean and variance can be obtained using the formulas that

$∑ i = 1 k i = k ( k + 1 ) 2 and ∑ i = 1 k i 2 = k ( k + 1 ) ( 2 k + 1 ) 6 .$ Figure 3.1   The probability mass function when N = 10

#### 3.2  Moments

Mean:

$N + 1 2$

Variance:

$( N − 1 ) ( N + 1 ) 12$

Coefficient of Variation:

$( N − 1 3 ( N + 1 ) ) 1 2$

Coefficient of Skewness:

0

Coefficient of Kurtosis:

$3 − 6 ( N 2 + 1 ) 5 ( N − 1 ) ( N + 1 )$

Moment Generating Function:

$M X ( t ) = e t ( 1 − e N t ) N ( 1 − e t )$

Mean Deviation:

${ N 2 − 1 4 N if N is odd, N 4 if N is even .$

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