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) |

(6) | |||

(7) |

The Moment-Generating Function is

(8) |

(9) |

(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) |

(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

1999-05-26