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
With
and
, these can be written
The Characteristic Function is
![\begin{displaymath}
\phi(t)={2\over ht}\sin({\textstyle{1\over 2}}ht)e^{imt},
\end{displaymath}](u_176.gif) |
(5) |
where
The Moment-Generating Function is
![\begin{displaymath}
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}](u_181.gif) |
(8) |
so
![\begin{displaymath}
M(t) = \cases{
{e^{tb}-e^{ta}\over t(b-a)} & for $t \not = 0$\cr
0 & for $t = 0$,\cr}
\end{displaymath}](u_182.gif) |
(9) |
and
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
The Moments about the Mean are
so the Mean, Variance, Skewness, and Kurtosis are
The probability distribution function and cumulative distributions function for a discrete uniform
distribution are
for
, ...,
. The Moment-Generating Function is
The Moments about 0 are
![\begin{displaymath}
\mu'_m={1\over N} \sum_{n=1}^N n^m,
\end{displaymath}](u_216.gif) |
(26) |
so
and the Moments about the Mean are
The Mean, Variance, Skewness, and Kurtosis are
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 531 and 533, 1987.
© 1996-9 Eric W. Weisstein
1999-05-26