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Properties of PH distributions

Below, we summarize some of the basic properties of the PH distribution. First, the set of PH distributions is quite broad and, in theory, any nonnegative distribution can be approximated arbitrarily closely by a PH distribution.

Proposition 2   [132] The set of PH distributions is dense in the set of nonnegative distributions (distributions with support on ).

Observe that Proposition 1 follows immediately from Proposition 2.

Second, the set of PH distributions is closed under some operations. In particular, a mixture of independent PH distributions is a PH distribution, and the convolution of independent PH distributions is a PH distribution.

Proposition 3   [111] Consider two PH distributions: PH( ) with distribution function and PH( ) with distribution function . A mixture of the two PH distribution, which has distribution function , is a PH distribution, PH( ), where

Here, denotes a zero matrix.

The convolution of the two PH distributions, PH( ) and PH( ), is a PH distribution, PH( ), where

Here, and , where is a column vector of 1's.

To shed light on the expression , consider a random variable whose distribution function is and a random variable whose distribution function is . Then, random variable

has distribution function . Below, unless otherwise stated, we denote the (cumulative) distribution function of a distribution, , by .

Definition 9   Let be a random variable having a distribution . We denote the cumulative distribution function by , namely

Finally, the distribution function, the density function, the moments, and the Laplace transform of a PH distribution have simple mathematical expressions.

Proposition 4   [111] The distribution function of PH( ) is given by

for , where the matrix exponential is defined by . The density function of PH( ) is given by

for , where .

Let be a random variable with the PH( ) distribution. Then,

for . The Laplace transform of PH( ) is given by

where and is an identity matrix.

Next: State of the art Up: Brief tutorial on phase Previous: Subclasses of PH distribution   Contents
Takayuki Osogami 2005-07-19