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
Adobe ISBN:

10.1201/b19191-5

 

Abstract

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

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Discrete Uniform Distribution

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 .
The probability mass function when

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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