butools.ph.APH2ndMomentLowerBound
=================================

.. currentmodule:: butools.ph

.. np:function:: APH2ndMomentLowerBound

    .. list-table::
        :widths: 25 150

        * - Matlab:
          - :code:`m2 = APH2ndMomentLowerBound(m1, n)`
        * - Mathematica:
          - :code:`m2 = APH2ndMomentLowerBound[m1, n]`
        * - Python/Numpy:
          - :code:`m2 = APH2ndMomentLowerBound(m1, n)`

    Returns the lower bound of the second moment of acyclic 
    phase-type (APH) distributions of order n.

    Parameters
    ----------
    m1 : double
        The first moment
    n : int
        Number of states

    Returns
    -------
    m2 : double
        The lowest second moment an order-n APH can have with
        the given first moment.

    References
    ----------
    .. [1]  M. Telek and A. Heindl, "Moment bounds for acyclic 
            discrete and continuous phase-type distributions of
            second order," in In Proc. of UK Performance 
            Evaluation Workshop, UKPEW, 2002"

    Examples
    ========
    For Matlab:

    >>> mean = 1.9;
    >>> n = 4;
    >>> mom2 = APH2ndMomentLowerBound(mean, n);
    >>> disp(mom2);
           4.5125
    >>> cv2 = mom2/mean^2-1;
    >>> disp(1/cv2);
         4

    For Mathematica:

    >>> mean = 1.9;
    >>> n = 4;
    >>> mom2 = APH2ndMomentLowerBound[mean, n];
    >>> Print[mom2];
    4.5125
    >>> cv2 = mom2/mean^2-1;
    >>> Print[1/cv2];
    3.9999999999999964

    For Python/Numpy:

    >>> mean = 1.9
    >>> n = 4
    >>> mom2 = APH2ndMomentLowerBound(mean, n)
    >>> print(mom2)
    4.5125
    >>> cv2 = mom2/mean**2-1
    >>> print(1/cv2)
    4.0