discrete uniform distribution

Key facts

Notation

U { a , b } {\displaystyle {\mathcal {U}}\{a,b\}} or u n i f { a , b } {\displaystyle \mathrm {unif} \{a,b\}}

Parameters

a , b {\displaystyle a,b} integers with b ≥ a {\displaystyle b\geq a} , n = b − a + 1 {\displaystyle n=b-a+1}

Support

k ∈ { a , a + 1 , … , b − 1 , b } {\displaystyle k\in \{a,a+1,\dots ,b-1,b\}}

Pmf

1 n {\displaystyle {\frac {1}{n}}}

Cdf

⌊ k ⌋ − a + 1 n {\displaystyle {\frac {\lfloor k\rfloor -a+1}{n}}}

Mean

a + b 2 {\displaystyle {\frac {a+b}{2}}}

Median

a + b 2 {\displaystyle {\frac {a+b}{2}}}

Mode

N/A

Variance

( b − a + 1 ) 2 − 1 12 {\displaystyle {\frac {(b-a+1)^{2}-1}{12}}}

Skewness

0 {\displaystyle 0}

Excess kurtosis

− 6 ( n 2 + 1 ) 5 ( n 2 − 1 ) {\displaystyle -{\frac {6(n^{2}+1)}{5(n^{2}-1)}}}

Entropy

ln ⁡ ( n ) {\displaystyle \ln(n)}

Mgf

e a t − e ( b + 1 ) t n ( 1 − e t ) {\displaystyle {\frac {e^{at}-e^{(b+1)t}}{n(1-e^{t})}}}

Cf

e i a t − e i ( b + 1 ) t n ( 1 − e i t ) {\displaystyle {\frac {e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}}}

Pgf

z a − z b + 1 n ( 1 − z ) {\displaystyle {\frac {z^{a}-z^{b+1}}{n(1-z)}}}

via Wikipedia infobox

Wikidata facts

via Wikidata · CC0