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Uniform Distribution

A distribution which has constant probability is called a uniform distribution, sometimes also called a Rectangular Distribution. The probability density function and cumulative distribution function for a continuous uniform distribution are

$\displaystyle P(x)$ $\textstyle =$ $\displaystyle \left\{\begin{array}{ll} {1\over b-a} & \mbox{for $a<x<b$}\\  0 & \mbox{for $x<a$, $x>b$}\end{array}\right.$ (1)
$\displaystyle D(x)$ $\textstyle =$ $\displaystyle \left\{\begin{array}{ll} 0 & \mbox{for $x < a$}\\  {x-a\over b-a} & \mbox{for $a \leq x < b$}\\  1 & \mbox{for $x \geq b$.}\end{array}\right.$ (2)

With $a=0$ and $b=1$, these can be written
$\displaystyle P(x)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\mathop{\rm sgn}\nolimits (x)-\mathop{\rm sgn}\nolimits (x-1)$ (3)
$\displaystyle D(x)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[1-(1-x)^2\mathop{\rm sgn}\nolimits (1-x)+x\mathop{\rm sgn}\nolimits (x)].$ (4)

The Characteristic Function is
\phi(t)={2\over ht}\sin({\textstyle{1\over 2}}ht)e^{imt},
\end{displaymath} (5)

$\displaystyle a$ $\textstyle =$ $\displaystyle m-{\textstyle{1\over 2}}h$ (6)
$\displaystyle b$ $\textstyle =$ $\displaystyle m+{\textstyle{1\over 2}}h.$ (7)

The Moment-Generating Function is
M(t)=\left\langle{e^{xt}}\right\rangle{} =\int_a^b {e^{xt}\over b-a} dx=\left[{e^{xt}\over t(b-a)}\right]_a^b,
\end{displaymath} (8)

M(t) = \cases{
{e^{tb}-e^{ta}\over t(b-a)} & for $t \not = 0$\cr
0 & for $t = 0$,\cr}
\end{displaymath} (9)

$\displaystyle M'(t)$ $\textstyle =$ $\displaystyle {1\over b-a}\left[{{1\over t}(be^{bt}-ae^{at})-{1\over t^2}(e^{bt}-e^{at})}\right]$  
  $\textstyle =$ $\displaystyle {e^{bt}(bt-1)-e^{at}(at-1)\over (b-a)t^2}.$ (10)

The function is not differentiable at zero, so the Moments cannot be found using the standard technique. They can, however, be found by direct integration. The Moments about 0 are
$\displaystyle \mu_1'$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(a+b)$ (11)
$\displaystyle \mu_2'$ $\textstyle =$ $\displaystyle {\textstyle{1\over 3}}(a^2+ab+b^2)$ (12)
$\displaystyle \mu_3'$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}(a+b)(a^2+b^2)$ (13)
$\displaystyle \mu_4'$ $\textstyle =$ $\displaystyle {\textstyle{1\over 5}}(a^4+a^3b+a^2b^2+ab^3+b^4).$ (14)

The Moments about the Mean are
$\displaystyle \mu_1$ $\textstyle =$ $\displaystyle 0$ (15)
$\displaystyle \mu_2$ $\textstyle =$ $\displaystyle {\textstyle{1\over 12}}(b-a)^2$ (16)
$\displaystyle \mu_3$ $\textstyle =$ $\displaystyle 0$ (17)
$\displaystyle \mu_4$ $\textstyle =$ $\displaystyle {\textstyle{1\over 80}}(b-a)^4,$ (18)

so the Mean, Variance, Skewness, and Kurtosis are
$\displaystyle \mu$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(a+b)$ (19)
$\displaystyle \sigma^2$ $\textstyle =$ $\displaystyle \mu_2={\textstyle{1\over 12}}(b-a)^2$ (20)
$\displaystyle \gamma_1$ $\textstyle =$ $\displaystyle {\mu_3\over\sigma^{3/2}}=0$ (21)
$\displaystyle \gamma_2$ $\textstyle =$ $\displaystyle -{\textstyle{6\over 5}}.$ (22)

The probability distribution function and cumulative distributions function for a discrete uniform distribution are

$\displaystyle P(n)$ $\textstyle =$ $\displaystyle {1\over N}$ (23)
$\displaystyle D(n)$ $\textstyle =$ $\displaystyle {n\over N}$ (24)

for $n=1$, ..., $N$. The Moment-Generating Function is
$\displaystyle M(t)$ $\textstyle =$ $\displaystyle \left\langle{e^{nt}}\right\rangle{}=\sum_{n=1}^N {1\over N} e^{nt} = {1\over N} {e^t-e^{t(N+1)}\over 1-e^t}$  
  $\textstyle =$ $\displaystyle {e^t(1-e^{Nt})\over N(1-e^t)}.$ (25)

The Moments about 0 are
\mu'_m={1\over N} \sum_{n=1}^N n^m,
\end{displaymath} (26)

$\displaystyle \mu_1'$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(N+1)$ (27)
$\displaystyle \mu_2'$ $\textstyle =$ $\displaystyle {\textstyle{1\over 6}}(N+1)(2N+1)$ (28)
$\displaystyle \mu_3'$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}N(N+1)^2$ (29)
$\displaystyle \mu_4'$ $\textstyle =$ $\displaystyle {\textstyle{1\over 30}}(N+1)(2N+1)(3N^2+3N-1),$ (30)

and the Moments about the Mean are
$\displaystyle \mu_2$ $\textstyle =$ $\displaystyle {\textstyle{1\over 12}}(N-1)(N+1)$ (31)
$\displaystyle \mu_3$ $\textstyle =$ $\displaystyle 0$ (32)
$\displaystyle \mu_4$ $\textstyle =$ $\displaystyle {\textstyle{1\over 240}}(N-1)(N+1)(3N^2-7).$ (33)

The Mean, Variance, Skewness, and Kurtosis are
$\displaystyle \mu$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(N+1)$ (34)
$\displaystyle \sigma^2$ $\textstyle =$ $\displaystyle \mu_2={\textstyle{1\over 12}}(N-1)(N+1)$ (35)
$\displaystyle \gamma_1$ $\textstyle =$ $\displaystyle {\mu_3\over\sigma^{3/2}}=0$ (36)
$\displaystyle \gamma_2$ $\textstyle =$ $\displaystyle {6(N^2+1)\over 5(N-1)(N+1)}.$ (37)


Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 531 and 533, 1987.

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© 1996-9 Eric W. Weisstein