Concluding remarks

Theorem 3 implies that the parameter $\lambda_Y$ of the EC distribution can be fixed without excluding the distributions of lowest variability from the set of EC distributions. In the rest of the chapter, we constrain $\lambda_Y$ as follows:

$$\lambda_Y = \frac{1}{(m_2^{X}-1)\mu_1^{X}},$$ (2.3)

and derive closed form representations of the remaining free parameters ($n$, $p$, $\lambda_{X1}$, $\lambda_{X2}$, $p_X$), where these free parameters will determine $m_2^{X}$ and $\mu_1^{X}$, which in turn gives $\lambda_Y$ by (2.3). Obviously, at least three degrees of freedom are necessary to match three moments. As we will see, the additional degrees of freedom allow us to accept all input distributions in ${\cal PH}_3$ and to use a smaller number of phases.