continuous uniform distribution

Key facts

Notation

U [ a , b ] {\displaystyle {\mathcal {U}}_{[a,b]}}

Parameters

− ∞ < a < b < ∞ {\displaystyle -\infty <a<b<\infty }

Support

[ a , b ] {\displaystyle [a,b]}

Pdf

{ 1 b − a for x ∈ [ a , b ] 0 otherwise {\displaystyle {\begin{cases}{\frac {1}{b-a}}&{\text{for }}x\in [a,b]\\0&{\text{otherwise}}\end{cases}}}

Cdf

b\n \\end{cases}"}}'> { 0 for x < a x − a b − a for x ∈ [ a , b ] 1 for x > b {\displaystyle {\begin{cases}0&{\text{for }}x<a\\{\frac {x-a}{b-a}}&{\text{for }}x\in [a,b]\\1&{\text{for }}x>b\end{cases}}}

Mean

1 2 ( a + b ) {\displaystyle {\tfrac {1}{2}}(a+b)}

Median

1 2 ( a + b ) {\displaystyle {\tfrac {1}{2}}(a+b)}

Mode

any value in ( a , b ) {\displaystyle {\text{any value in }}(a,b)}

Variance

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

Mad

1 4 ( b − a ) {\displaystyle {\tfrac {1}{4}}(b-a)}

Skewness

0 {\displaystyle 0}

Excess kurtosis

− 6 5 {\displaystyle -{\tfrac {6}{5}}}

Entropy

log ⁡ ( b − a ) {\displaystyle \log(b-a)}

Mgf

{ e t b − e t a t ( b − a ) for t ≠ 0 1 for t = 0 {\displaystyle {\begin{cases}{\frac {\mathrm {e} ^{tb}-\mathrm {e} ^{ta}}{t(b-a)}}&{\text{for }}t\neq 0\\1&{\text{for }}t=0\end{cases}}}

Cf

{ e i t b − e i t a i t ( b − a ) for t ≠ 0 1 for t = 0 {\displaystyle {\begin{cases}{\frac {\mathrm {e} ^{\mathrm {i} tb}-\mathrm {e} ^{\mathrm {i} ta}}{\mathrm {i} t(b-a)}}&{\text{for }}t\neq 0\\1&{\text{for }}t=0\end{cases}}}

via Wikipedia infobox