EC distribution

To motivate the theorem in this section, suppose one is trying to match the first three moments of a given distribution, , to a distribution, , which is the convolution of exponential distributions (possibly with different rates) and a two-phase Coxian PH distribution. If has sufficiently high second and third moments, then a two-phase Coxian PH distribution alone suffices and we need no exponential distributions (recall Theorem 2). If the variability of is lower, however, we might try appending an exponential distribution to the two-phase Coxian PH distribution. If that does not suffice, we might append two exponential distributions to the two-phase Coxian PH distribution. Thus, if has very low variability, we might be forced to use many phases to get the variability of to be low enough. Therefore, to minimize the number of phases in , it seems desirable to choose the rates of the exponential distributions so that the overall variability of is minimized. One could express the appending of each exponential distribution as a ``function'' whose goal is to reduce the variability of yet further.

In theory, function
allows each successive exponential distribution which
is appended to have a different rate.
Surprisingly, however, the following theorem shows that if the
exponential distribution being appended by function is chosen
so as to minimize the normalized second moment of (as specified
by the definition), then
the rate of each successive is always *the same* and is
defined by the simple
formula shown in the theorem below.
The theorem also characterizes the normalized
moments of .

**Proof**:__We first characterize
__, where is an arbitrary distribution with a finite
third moment and is an exponential distribution.
The normalized second moment of is

where . Observe that is minimized when , namely when

Observe that when is set at this value, the normalized moments of satisfy:

__We next characterize
for .__
By the above expression on and , the second part of the theorem on the normalized moments of
follow from solving the following recurrence equations (where we use
to denote
and to denote
):

The solutions for these recurrence equations are

for all . These solutions can be verified by substitution. This completes the proof of the second part of the theorem.

The first part of the theorem on
is proved by induction.
When ,
(2.1)
follows from (2.2).
Assume that
(2.1)
holds for .
Let
.
By the second part of the theorem, which is proved above,

Thus, by (2.2),

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**Proof**:By Theorem 3, is a continuous and
monotonically increasing function of . Thus, the infimum and
the supremum of are given by evaluating at the infimum
and the supremum, respectively, of . When
,
. When
,
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Corollary 1 suggests the number, , of times that function must be applied to to bring into a desired range, given the value of .