## 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

 (1) (2)

With and , these can be written
 (3) (4)

The Characteristic Function is
 (5)

where
 (6) (7)

The Moment-Generating Function is
 (8)

so
 (9)

and
 (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
 (11) (12) (13) (14)

The Moments about the Mean are
 (15) (16) (17) (18)

so the Mean, Variance, Skewness, and Kurtosis are
 (19) (20) (21) (22)

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

 (23) (24)

for , ..., . The Moment-Generating Function is
 (25)

The Moments about 0 are
 (26)

so
 (27) (28) (29) (30)

and the Moments about the Mean are
 (31) (32) (33)

The Mean, Variance, Skewness, and Kurtosis are
 (34) (35) (36) (37)

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