Special Cases
The following probability distributions are all considered special cases of a continuous phase-type distribution:
- Degenerate distribution, point mass at zero or the empty phase-type distribution - 0 phases.
- Exponential distribution - 1 phase.
- Erlang distribution - 2 or more identical phases in sequence.
- Deterministic distribution (or constant) - The limiting case of an Erlang distribution, as the number of phases become infinite, while the time in each state becomes zero.
- Coxian distribution - 2 or more (not necessarily identical) phases in sequence, with a probability of transitioning to the terminating/absorbing state after each phase.
- Hyper-exponential distribution (also called a mixture of exponential) - 2 or more non-identical phases, that each have a probability of occurring in a mutually exclusive, or parallel, manner. (Note: The exponential distribution is the degenerate situation when all the parallel phases are identical.)
- Hypoexponential distribution - 2 or more phases in sequence, can be non-identical or a mixture of identical and non-identical phases, generalises the Erlang.
As the phase-type distribution is dense in the field of all positive-valued distributions, we can represent any positive valued distribution. However, the phase-type is a light-tailed or platikurtic distribution. So the representation of heavy-tailed or leptokurtic distribution by phase type is an approximation, even if the precision of the approximation can be as good as we want.
Read more about this topic: Phase-type Distribution
Famous quotes containing the words special and/or cases:
“When a mother quarrels with a daughter, she has a double dose of unhappiness—hers from the conflict, and empathy with her daughter’s from the conflict with her. Throughout her life a mother retains this special need to maintain a good relationship with her daughter.”
—Terri Apter (20th century)
“Lovers’ quarrels are not generally about money. Divorce cases generally are.”
—Mason Cooley (b. 1927)