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.. |cr| raw:: latex

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.. _rst_dist_rice: 

Rice (Nakagami-n) distribution
-------------------------------------------------------------------------------


.. py:class:: ctx.dist_rice(n1, n2, lambda, **kwargs)

    These functions return PDF, CDF, and ICDF of the Rice distribution with location
    `a`, scale `b > 0`, and the support interval `(-\infty,+\infty)`.

    See also: Wikipedia :cite:p:`WikipediaDis84`, :cite:t:`CharfunDis84`.




|cr|

.. method:: dist_rice.pdf(x)

    Returns `\text{pdf}_X(x)`, the probability density function (pdf) of a random variable `X`, following an Rice distribution:

    .. math:: \text{pdf}_X(x) = \frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)}{2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)



    .. code-block:: python

        >>> from mpdistrib import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; x = 3; 
        >>> print ("pdf: ", dist_rice(mu, sigma).pdf(x))
        6.3563523462564525615615615614561356E-20



|cr|


.. method:: dist_rice.cdf(x)

    Returns `\text{cdf}_X(x)`, the cumulative distribution function (cdf) of a random variable `X`, following an Rice distribution:

    .. math:: \text{cdf}_X(x) = 1-Q_1\left(\frac{\nu}{\sigma },\frac{x}{\sigma }\right)



    .. code-block:: python

        >>> from mpdistrib import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; x = 3; 
        >>> print ("cdf: ", dist_rice(mu, sigma).pdf(x))
        6.3563523462564525615615615614561356E-20




|cr|

.. method:: dist_rice.sf(x)

    Returns `\text{sf}_X(x)`, the survival function function (sf) of a random variable `X`, following an Rice distribution:


    .. math:: \text{sf}_X(x) = 1 - \text{cdf}_X(x).


    .. code-block:: python

        >>> from mpdistrib import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; x = 3; 
        >>> print (" sf: ", dist_rice(mu, sigma).pdf(x))
        sf: 6.3563523462564525615615615614561356E-20



|cr|

.. method:: dist_rice.qtf(q)

    Returns `\text{qtf}_X(x)`, the quantile function function (qtf) of a random variable `X`, following an Rice distribution:

    .. math:: \text{qtf}_X(q) = ??



    .. code-block:: python

        >>> from mpdistrib import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; q = 0.3; 
        >>> print ("qtf: ", dist_rice(mu, sigma).qtf(q))
        qtf: 6.3563523462564525615615615614561356E+00




|cr|

.. method:: dist_rice.isf(q)

    Returns `\text{isf}_X(q)`, the inverse survival function function (isf) of a random variable `X`, following an Rice distribution:

    .. math:: \text{isf}_X(q) = ??


    .. code-block:: python

        >>> from mpdistrib import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; q = 0.3; 
        >>> print ("isf: ", dist_rice(mu, sigma).isf(q))
        6.3563523462564525615615615614561356E+00




|cr|

.. method:: dist_rice.c_x(t)

    Returns `C_X(t)`, the characteristic function of a random variable `X`, following an Rice distribution:

    .. math::  C_X(t) = ??


    .. code-block:: python

        >>> from mpdistrib import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; t = 0.3; 
        >>> print ("c_x: ", dist_rice(mu, sigma).c_x(t))
        6.3563523462564525615615615614561356E+00




|cr|

.. method:: dist_rice.m_x(t)

    Returns `M_X(t)`, the moment generating function of a random variable `X`, following an Rice distribution:

    .. math:: M_X(t) = ??



    .. code-block:: python

        >>> from mpdistrib import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; t = 0.3; 
        >>> print ("m_x: ", dist_rice(mu, sigma).c_x(t))
        6.3563523462564525615615615614561356E+00




|cr|

.. method:: dist_rice.k_x(t, k = 0)

    Returns `K_X(t)`, the cumulant generating function of a random variable `X`, following an Rice distribution:

    .. math:: K_X(t) = ??


    .. code-block:: python

        >>> from mpdistrib import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; t = 0.3; k = 6;
        >>> print ("c_x: ", dist_rice(mu, sigma).k_x(t, k))
        6.3563523462564525615615615614561356E+00








|cr|

.. method:: dist_rice.moments(k)

    Returns the first `j` central moments, `\mu_j, j = 1 \ldots k`, of a random variable `X`, 
    following an Rice distribution (Wikipedia). The raw moments are calculated from the central moments.


    .. math::  \mu_{X}(r) =  \sigma^k2^{k/2}\,\Gamma(1\!+\!k/2)\,L_{k/2}(-\nu^2/2\sigma^2),

    .. math::  L_q(x)=L_q^{(0)}(x)=M(-q,1,x)=\,_1F_1(-q;1;x)

    .. code-block:: python

        >>> from mpdistrib import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; k = 6;
        >>> print ("saddlepoint: ", dist_rice(mu, sigma).moments(k))
        6.3563523462564525615615615614561356E+00



|cr|

.. method:: dist_rice.cumulants(k)

    Returns the first `j` cumulants, `\kappa_j, j = 1 \ldots k`, of a random variable `X`,  following an Rice distribution. The cumulants are calculated from the moments.


    .. code-block:: python

        >>> from mpdistrib import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; k = 6;
        >>> print ("saddlepoint: ", dist_rice(mu, sigma).cumulants(k))
        6.3563523462564525615615615614561356E+00