.. code to create: see blog/agg/examples/probems_and_solutions.ipynb

.. _list of freq distributions:

Appendix: Selected ``scipy.stats`` Discrete Random Variables
-----------------------------------------------------------------

Here is the list of ``scipy.stats`` discrete random variables.

::

                             Num. args   Min range   Max range Parameters
    Distribution
    bernoulli                         1           0           1 'p'
    dlaplace                          1        -inf         inf 'a'
    geom                              1           1         inf 'p'
    logser                            1           1         inf 'p'
    planck                            1           0         inf 'lambda\_'
    poisson                           1           0         inf 'mu'
    yulesimon                         1           1         inf 'alpha'
    zipf                              1           1         inf 'a'
    binom                             2           0         inf 'n' and 'p'
    boltzmann                         2           0         inf 'lambda\_' and 'N'
    nbinom                            2           0         inf 'n' and 'p'
    randint                           2           0         inf 'low' and 'high'
    skellam                           2        -inf         inf 'mu1' and 'mu2'
    zipfian                           2           1         inf 'a' and 'n'
    betabinom                         3           0         inf 'n', 'a', and 'b'
    hypergeom                         3           0         inf 'M', 'n', and 'N'
    nhypergeom                        3           0         inf 'M', 'n', and 'r'
    nchypergeom_fisher                4           0         inf 'M', 'n', 'N', and 'odds'
    nchypergeom_wallenius             4           0         inf 'M', 'n', 'N', and 'odds'


.. _freq dist bernoulli:

* ``bernoulli`` **Bernoulli** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.bernoulli.html>`_). The probability mass function for `bernoulli` is:

    .. math::

       f(k) = \begin{cases}1-p  &\text{if } k = 0\\
                           p    &\text{if } k = 1\end{cases}

    for :math:`k` in :math:`\{0, 1\}`, :math:`0 \leq p \leq 1`

    `bernoulli` takes :math:`p` as shape parameter,
    where :math:`p` is the probability of a single success
    and :math:`1-p` is the probability of a single failure.

    The probability mass function above is defined in the "standardized" form.
    To shift distribution use the ``loc`` parameter.
    Specifically, ``bernoulli.pmf(k, p, loc)`` is identically
    equivalent to ``bernoulli.pmf(k - loc, p)``.


.. _freq dist betabinom:

* ``betabinom`` **Betabinom** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.betabinom.html>`_). The beta-binomial distribution is a binomial distribution with a
    probability of success `p` that follows a beta distribution.

    The probability mass function for `betabinom` is:

    .. math::

       f(k) = \binom{n}{k} \frac{B(k + a, n - k + b)}{B(a, b)}

    for :math:`k \in \{0, 1, \dots, n\}`, :math:`n \geq 0`, :math:`a > 0`,
    :math:`b > 0`, where :math:`B(a, b)` is the beta function.

    `betabinom` takes :math:`n`, :math:`a`, and :math:`b` as shape parameters.


.. _freq dist binom:

* ``binom`` **Binom** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.binom.html>`_). The probability mass function for `binom` is:

    .. math::

       f(k) = \binom{n}{k} p^k (1-p)^{n-k}

    for :math:`k \in \{0, 1, \dots, n\}`, :math:`0 \leq p \leq 1`

    `binom` takes :math:`n` and :math:`p` as shape parameters,
    where :math:`p` is the probability of a single success
    and :math:`1-p` is the probability of a single failure.


.. _freq dist boltzmann:

* ``boltzmann`` **Boltzmann** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.boltzmann.html>`_). The probability mass function for `boltzmann` is:

    .. math::

        f(k) = (1-\exp(-\lambda)) \exp(-\lambda k) / (1-\exp(-\lambda N))

    for :math:`k = 0,..., N-1`.

    `boltzmann` takes :math:`\lambda > 0` and :math:`N > 0` as shape parameters.



.. _freq dist geom:

* ``geom`` **Geom** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.geom.html>`_). The probability mass function for `geom` is:

    .. math::

        f(k) = (1-p)^{k-1} p

    for :math:`k \ge 1`, :math:`0 < p \leq 1`

    `geom` takes :math:`p` as shape parameter,
    where :math:`p` is the probability of a single success
    and :math:`1-p` is the probability of a single failure.


.. _freq dist logser:

* ``logser`` **Logser** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.logser.html>`_). The probability mass function for `logser` is:

    .. math::

        f(k) = - \frac{p^k}{k \log(1-p)}

    for :math:`k \ge 1`, :math:`0 < p < 1`

    `logser` takes :math:`p` as shape parameter,
    where :math:`p` is the probability of a single success
    and :math:`1-p` is the probability of a single failure.


.. _freq dist nbinom:

* ``nbinom`` **Nbinom** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.nbinom.html>`_). Negative binomial distribution describes a sequence of i.i.d. Bernoulli
    trials, repeated until a predefined, non-random number of successes occurs.

    The probability mass function of the number of failures for `nbinom` is:

    .. math::

       f(k) = \binom{k+n-1}{n-1} p^n (1-p)^k

    for :math:`k \ge 0`, :math:`0 < p \leq 1`

    `nbinom` takes :math:`n` and :math:`p` as shape parameters where n is the
    number of successes, :math:`p` is the probability of a single success,
    and :math:`1-p` is the probability of a single failure.

    Another common parameterization of the negative binomial distribution is
    in terms of the mean number of failures :math:`\mu` to achieve :math:`n`
    successes. The mean :math:`\mu` is related to the probability of success
    as

    .. math::

       p = \frac{n}{n + \mu}

    The number of successes :math:`n` may also be specified in terms of a
    "dispersion", "heterogeneity", or "aggregation" parameter :math:`\alpha`,
    which relates the mean :math:`\mu` to the variance :math:`\sigma^2`,
    e.g. :math:`\sigma^2 = \mu + \alpha \mu^2`. Regardless of the convention
    used for :math:`\alpha`,

    .. math::

       p &= \frac{\mu}{\sigma^2} \\
       n &= \frac{\mu^2}{\sigma^2 - \mu}


.. _freq dist planck:

* ``planck`` **Planck** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.planck.html>`_). The probability mass function for `planck` is:

    .. math::

        f(k) = (1-\exp(-\lambda)) \exp(-\lambda k)

    for :math:`k \ge 0` and :math:`\lambda > 0`.

    `planck` takes :math:`\lambda` as shape parameter. The Planck distribution
    can be written as a geometric distribution (`geom`) with
    :math:`p = 1 - \exp(-\lambda)` shifted by ``loc = -1``.



.. _freq dist poisson:

* ``poisson`` **Poisson** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html>`_). The probability mass function for `poisson` is:

    .. math::

        f(k) = \exp(-\mu) \frac{\mu^k}{k!}

    for :math:`k \ge 0`.

    `poisson` takes :math:`\mu \geq 0` as shape parameter.
    When :math:`\mu = 0`, the ``pmf`` method
    returns ``1.0`` at quantile :math:`k = 0`.


.. _freq dist randint:

* ``randint`` **Randint** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.randint.html>`_). The probability mass function for `randint` is:

    .. math::

        f(k) = \frac{1}{\texttt{high} - \texttt{low}}

    for :math:`k \in \{\texttt{low}, \dots, \texttt{high} - 1\}`.

    `randint` takes :math:`\texttt{low}` and :math:`\texttt{high}` as shape
    parameters.


.. _list of distributions:

Appendix: ``scipy.stats`` Continuous Random Variables
------------------------------------------------------

The information below was extracted from the `scipy help for continuous distributions <https://docs.scipy.org/doc/scipy/reference/stats.html#continuous-distributions>`_. The basic list can be created by introspection---wonderful Python!


.. ipython:: python
    :okwarning:

    import scipy.stats as ss
    import pandas as pd

    ans = []
    for k in dir(ss):
        ob = getattr(ss, k)
        if str(type(ob)).find('continuous_distns') > 0:
            try:
                fz = ob()
            except TypeError as e:
                ee = e
                ans.append([k, str(e), -1, ob.a, ob.b])
            else:
                ans.append([k, 'no args fine', 0, ob.a, ob.b])

    df = pd.DataFrame(ans, columns=['dist', 'm', 'args', 'a', 'b'])
    for i in range(1,5):
        df.loc[df.m.str.find(f'{i} required')>=0, 'args'] = i

    df = df.sort_values(['args', 'dist'])
    df['params'] = ''
    df.loc[df.args > 0, 'params'] = df.loc[df.args > 0, 'm'].str.split(':').str[1]
    df = df.drop(columns='m')

    print(df.rename(columns={'dist': 'Distribution', 'args': 'Num. args',
            'a': 'Min range' , 'b': 'Max range', 'params': 'Parameters'}).\
            set_index('Distribution').to_string(float_format=lambda x: f'{x:.4g}'))


.. _dist alpha:

* ``alpha`` **Alpha** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.alpha.html>`_). The probability density function for `alpha` is:

    .. math::

        f(x, a) = \frac{1}{x^2 \Phi(a) \sqrt{2\pi}} *
                  \exp(-\frac{1}{2} (a-1/x)^2)

    where :math:`\Phi` is the normal CDF, :math:`x > 0`, and :math:`a > 0`.

    `alpha` takes ``a`` as a shape parameter.


.. _dist anglit:

* ``anglit`` **Anglit** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.anglit.html>`_). The probability density function for `anglit` is:

    .. math::

        f(x) = \sin(2x + \pi/2) = \cos(2x)

    for :math:`-\pi/4 \le x \le \pi/4`.


.. _dist arcsine:

* ``arcsine`` **Arcsine** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.arcsine.html>`_). The probability density function for `arcsine` is:

    .. math::

        f(x) = \frac{1}{\pi \sqrt{x (1-x)}}

    for :math:`0 < x < 1`.


.. _dist argus:

* ``argus`` **Argus** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.argus.html>`_). The probability density function for `argus` is:

    .. math::

        f(x, \chi) = \frac{\chi^3}{\sqrt{2\pi} \Psi(\chi)} x \sqrt{1-x^2}
                     \exp(-\chi^2 (1 - x^2)/2)

    for :math:`0 < x < 1` and :math:`\chi > 0`, where

    .. math::

        \Psi(\chi) = \Phi(\chi) - \chi \phi(\chi) - 1/2

    with :math:`\Phi` and :math:`\phi` being the CDF and PDF of a standard
    normal distribution, respectively.

    `argus` takes :math:`\chi` as shape a parameter.


.. _dist beta:

* ``beta`` **Beta** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.beta.html>`_). The probability density function for `beta` is:

    .. math::

        f(x, a, b) = \frac{\Gamma(a+b) x^{a-1} (1-x)^{b-1}}
                          {\Gamma(a) \Gamma(b)}

    for :math:`0 <= x <= 1`, :math:`a > 0`, :math:`b > 0`, where
    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

    `beta` takes :math:`a` and :math:`b` as shape parameters.


.. _dist betaprime:

* ``betaprime`` **Beta Prime** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.betaprime.html>`_). The probability density function for `betaprime` is:

    .. math::

        f(x, a, b) = \frac{x^{a-1} (1+x)^{-a-b}}{\beta(a, b)}

    for :math:`x >= 0`, :math:`a > 0`, :math:`b > 0`, where
    :math:`\beta(a, b)` is the beta function (see `scipy.special.beta`).

    `betaprime` takes ``a`` and ``b`` as shape parameters.


.. _dist bradford:

* ``bradford`` **Bradford** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.bradford.html>`_). The probability density function for `bradford` is:

    .. math::

        f(x, c) = \frac{c}{\log(1+c) (1+cx)}

    for :math:`0 <= x <= 1` and :math:`c > 0`.

    `bradford` takes ``c`` as a shape parameter for :math:`c`.


.. _dist burr:

* ``burr`` **Burr (Type III)** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.burr.html>`_). The probability density function for `burr` is:

    .. math::

        f(x, c, d) = c d x^{-c - 1} / (1 + x^{-c})^{d + 1}

    for :math:`x >= 0` and :math:`c, d > 0`.

    `burr` takes :math:`c` and :math:`d` as shape parameters.

    This is the PDF corresponding to the third CDF given in Burr's list;
    specifically, it is equation (11) in Burr's paper. The distribution
    is also commonly referred to as the Dagum distribution. If the
    parameter :math:`c < 1` then the mean of the distribution does not
    exist and if :math:`c < 2` the variance does not exist.
    The PDF is finite at the left endpoint :math:`x = 0` if :math:`c * d >= 1`.


.. _dist burr12:

* ``burr12`` **Burr (Type XII)** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.burr12.html>`_). The probability density function for `burr` is:

    .. math::

        f(x, c, d) = c d x^{c-1} / (1 + x^c)^{d + 1}

    for :math:`x >= 0` and :math:`c, d > 0`.

    `burr12` takes ``c`` and ``d`` as shape parameters for :math:`c`
    and :math:`d`.

    This is the PDF corresponding to the twelfth CDF given in Burr's list;
    specifically, it is equation (20) in Burr's paper.


.. _dist cauchy:

* ``cauchy`` **Cauchy** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.cauchy.html>`_). The probability density function for `cauchy` is

    .. math::

        f(x) = \frac{1}{\pi (1 + x^2)}

    for a real number :math:`x`.


.. _dist chi:

* ``chi`` **Chi** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.chi.html>`_). The probability density function for `chi` is:

    .. math::

        f(x, k) = \frac{1}{2^{k/2-1} \Gamma \left( k/2 \right)}
                   x^{k-1} \exp \left( -x^2/2 \right)

    for :math:`x >= 0` and :math:`k > 0` (degrees of freedom, denoted ``df``
    in the implementation). :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    Special cases of `chi` are:

        - ``chi(1, loc, scale)`` is equivalent to `halfnorm`
        - ``chi(2, 0, scale)`` is equivalent to `rayleigh`
        - ``chi(3, 0, scale)`` is equivalent to `maxwell`

    `chi` takes ``df`` as a shape parameter.


.. _dist chi2:

* ``chi2`` **Chi-squared** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.chi2.html>`_). The probability density function for `chi2` is:

    .. math::

        f(x, k) = \frac{1}{2^{k/2} \Gamma \left( k/2 \right)}
                   x^{k/2-1} \exp \left( -x/2 \right)

    for :math:`x > 0`  and :math:`k > 0` (degrees of freedom, denoted ``df``
    in the implementation).

    `chi2` takes ``df`` as a shape parameter.

    The chi-squared distribution is a special case of the gamma
    distribution, with gamma parameters ``a = df/2``, ``loc = 0`` and
    ``scale = 2``.


.. _dist cosine:

* ``cosine`` **Cosine** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.cosine.html>`_). The cosine distribution is an approximation to the normal distribution. The probability density function for `cosine` is:

    .. math::

        f(x) = \frac{1}{2\pi} (1+\cos(x))

    for :math:`-\pi \le x \le \pi`.


.. _dist crystalball:

* ``crystalball`` **Crystalball** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.crystalball.html>`_). The probability density function for `crystalball` is:

    .. math::

        f(x, \beta, m) =  \begin{cases}
                            N \exp(-x^2 / 2),  &\text{for } x > -\beta\\
                            N A (B - x)^{-m}  &\text{for } x \le -\beta
                          \end{cases}

    where :math:`A = (m / |\beta|)^m  \exp(-\beta^2 / 2)`,
    :math:`B = m/|\beta| - |\beta|` and :math:`N` is a normalisation constant.

    `crystalball` takes :math:`\beta > 0` and :math:`m > 1` as shape
    parameters.  :math:`\beta` defines the point where the pdf changes
    from a power-law to a Gaussian distribution.  :math:`m` is the power
    of the power-law tail.


.. _dist dgamma:

* ``dgamma`` **Double Gamma** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.dgamma.html>`_). The probability density function for `dgamma` is:

    .. math::

        f(x, a) = \frac{1}{2\Gamma(a)} |x|^{a-1} \exp(-|x|)

    for a real number :math:`x` and :math:`a > 0`. :math:`\Gamma` is the
    gamma function (`scipy.special.gamma`).

    `dgamma` takes ``a`` as a shape parameter for :math:`a`.


.. _dist dweibull:

* ``dweibull`` **Double Weibull** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.dweibull.html>`_). The probability density function for `dweibull` is given by

    .. math::

        f(x, c) = c / 2 |x|^{c-1} \exp(-|x|^c)

    for a real number :math:`x` and :math:`c > 0`.

    `dweibull` takes ``c`` as a shape parameter for :math:`c`.


.. _dist erlang:

* ``erlang`` **Erlang** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.erlang.html>`_). The Erlang distribution is a special case of the Gamma distribution, with the shape parameter `a` an integer.  Note that this restriction is not enforced by `erlang`. It will, however, generate a warning the first time a non-integer value is used for the shape parameter.

    :ref:`Refer to <dist gamma>` `gamma` for examples.


.. _dist expon:

* ``expon`` **Exponential** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.expon.html>`_). The probability density function for `expon` is:

    .. math::

        f(x) = \exp(-x)

    for :math:`x \ge 0`.


.. _dist exponnorm:

* ``exponnorm`` **Exponentially Modified Normal** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.exponnorm.html>`_). The probability density function for `exponnorm` is:

    .. math::

        f(x, K) = \frac{1}{2K} \exp\left(\frac{1}{2 K^2} - x / K \right)
                  \text{erfc}\left(-\frac{x - 1/K}{\sqrt{2}}\right)

    where :math:`x` is a real number and :math:`K > 0`.

    It can be thought of as the sum of a standard normal random variable
    and an independent exponentially distributed random variable with rate
    ``1/K``.


.. _dist exponweib:

* ``exponweib`` **Exponentiated Weibull** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.exponweib.html>`_). The probability density function for `exponweib` is:

    .. math::

        f(x, a, c) = a c [1-\exp(-x^c)]^{a-1} \exp(-x^c) x^{c-1}

    and its cumulative distribution function is:

    .. math::

        F(x, a, c) = [1-\exp(-x^c)]^a

    for :math:`x > 0`, :math:`a > 0`, :math:`c > 0`.

    `exponweib` takes :math:`a` and :math:`c` as shape parameters:

    * :math:`a` is the exponentiation parameter,
      with the special case :math:`a=1` corresponding to the
      (non-exponentiated) Weibull distribution `weibull_min`.
    * :math:`c` is the shape parameter of the non-exponentiated Weibull law.


.. _dist exponpow:

* ``exponpow`` **Exponential Power** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.exponpow.html>`_). The probability density function for `exponpow` is:

    .. math::

        f(x, b) = b x^{b-1} \exp(1 + x^b - \exp(x^b))

    for :math:`x \ge 0`, :math:`b > 0`.  Note that this is a different
    distribution from the exponential power distribution that is also known
    under the names "generalized normal" or "generalized Gaussian".

    `exponpow` takes ``b`` as a shape parameter for :math:`b`.


.. _dist f:

* ``f`` **F (Snecdor F)** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.f.html>`_). The probability density function for `f` is:

    .. math::

        f(x, df_1, df_2) = \frac{df_2^{df_2/2} df_1^{df_1/2} x^{df_1 / 2-1}}
                                {(df_2+df_1 x)^{(df_1+df_2)/2}
                                 B(df_1/2, df_2/2)}

    for :math:`x > 0`.

    `f` takes ``dfn`` and ``dfd`` as shape parameters.


.. _dist fatiguelife:

* ``fatiguelife`` **Fatigue Life (Birnbaum-Saunders)** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.fatiguelife.html>`_). The probability density function for `fatiguelife` is:

    .. math::

        f(x, c) = \frac{x+1}{2c\sqrt{2\pi x^3}} \exp(-\frac{(x-1)^2}{2x c^2})

    for :math:`x >= 0` and :math:`c > 0`.

    `fatiguelife` takes ``c`` as a shape parameter for :math:`c`.


.. _dist fisk:

* ``fisk`` **Fisk** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.fisk.html>`_). The probability density function for `fisk` is:

    .. math::

        f(x, c) = c x^{-c-1} (1 + x^{-c})^{-2}

    for :math:`x >= 0` and :math:`c > 0`.

    `fisk` takes ``c`` as a shape parameter for :math:`c`.

    `fisk` is a special case of `burr` or `burr12` with ``d=1``.


.. _dist foldcauchy:

* ``foldcauchy`` **Folded Cauchy** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.foldcauchy.html>`_). The probability density function for `foldcauchy` is:

    .. math::

        f(x, c) = \frac{1}{\pi (1+(x-c)^2)} + \frac{1}{\pi (1+(x+c)^2)}

    for :math:`x \ge 0`.

    `foldcauchy` takes ``c`` as a shape parameter for :math:`c`.


.. _dist foldnorm:

* ``foldnorm`` **Folded Normal** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.foldnorm.html>`_). The probability density function for `foldnorm` is:

    .. math::

        f(x, c) = \sqrt{2/\pi} cosh(c x) \exp(-\frac{x^2+c^2}{2})

    for :math:`c \ge 0`.

    `foldnorm` takes ``c`` as a shape parameter for :math:`c`.


.. _dist genlogistic:

* ``genlogistic`` **Generalized Logistic** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.genlogistic.html>`_). The probability density function for `genlogistic` is:

    .. math::

        f(x, c) = c \frac{\exp(-x)}
                         {(1 + \exp(-x))^{c+1}}

    for :math:`x >= 0`, :math:`c > 0`.

    `genlogistic` takes ``c`` as a shape parameter for :math:`c`.


.. _dist gennorm:

* ``gennorm`` **Generalized normal** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gennorm.html>`_). The probability density function for `gennorm` is:

    .. math::

        f(x, \beta) = \frac{\beta}{2 \Gamma(1/\beta)} \exp(-|x|^\beta)

    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

    `gennorm` takes ``beta`` as a shape parameter for :math:`\beta`.
    For :math:`\beta = 1`, it is identical to a Laplace distribution.
    For :math:`\beta = 2`, it is identical to a normal distribution
    (with ``scale=1/sqrt(2)``).


.. _dist genpareto:

* ``genpareto`` **Generalized Pareto** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.genpareto.html>`_). The probability density function for `genpareto` is:

    .. math::

        f(x, c) = (1 + c x)^{-1 - 1/c}

    defined for :math:`x \ge 0` if :math:`c \ge 0`, and for
    :math:`0 \le x \le -1/c` if :math:`c < 0`.

    `genpareto` takes ``c`` as a shape parameter for :math:`c`.

    For :math:`c=0`, `genpareto` reduces to the exponential
    distribution, `expon`:

    .. math::

        f(x, 0) = \exp(-x)

    For :math:`c=-1`, `genpareto` is uniform on ``[0, 1]``:

    .. math::

        f(x, -1) = 1


.. _dist genexpon:

* ``genexpon`` **Generalized Exponential** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.genexpon.html>`_). The probability density function for `genexpon` is:

    .. math::

        f(x, a, b, c) = (a + b (1 - \exp(-c x)))
                        \exp(-a x - b x + \frac{b}{c}  (1-\exp(-c x)))

    for :math:`x \ge 0`, :math:`a, b, c > 0`.

    `genexpon` takes :math:`a`, :math:`b` and :math:`c` as shape parameters.


.. _dist genextreme:

* ``genextreme`` **Generalized Extreme Value** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.genextreme.html>`_). For :math:`c=0`, `genextreme` is equal to `gumbel_r`. The probability density function for `genextreme` is:

    .. math::

        f(x, c) = \begin{cases}
                    \exp(-\exp(-x)) \exp(-x)              &\text{for } c = 0\\
                    \exp(-(1-c x)^{1/c}) (1-c x)^{1/c-1}  &\text{for }
                                                            x \le 1/c, c > 0
                  \end{cases}


    Note that several sources and software packages use the opposite
    convention for the sign of the shape parameter :math:`c`.

    `genextreme` takes ``c`` as a shape parameter for :math:`c`.


.. _dist gausshyper:

* ``gausshyper`` **Gauss Hypergeometric** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gausshyper.html>`_). The probability density function for `gausshyper` is:

    .. math::

        f(x, a, b, c, z) = C x^{a-1} (1-x)^{b-1} (1+zx)^{-c}

    for :math:`0 \le x \le 1`, :math:`a > 0`, :math:`b > 0`, :math:`z > -1`,
    and :math:`C = \frac{1}{B(a, b) F[2, 1](c, a; a+b; -z)}`.
    :math:`F[2, 1]` is the Gauss hypergeometric function
    `scipy.special.hyp2f1`.

    `gausshyper` takes :math:`a`, :math:`b`, :math:`c` and :math:`z` as shape
    parameters.


.. _dist gamma:

* ``gamma`` **Gamma** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gamma.html>`_). The probability density function for `gamma` is:

    .. math::

        f(x, a) = \frac{x^{a-1} e^{-x}}{\Gamma(a)}

    for :math:`x \ge 0`, :math:`a > 0`. Here :math:`\Gamma(a)` refers to the
    gamma function.

    `gamma` takes ``a`` as a shape parameter for :math:`a`.

    When :math:`a` is an integer, `gamma` reduces to the Erlang
    distribution, and when :math:`a=1` to the exponential distribution.

    Gamma distributions are sometimes parameterized with two variables,
    with a probability density function of:

    .. math::

        f(x, \alpha, \beta) = \frac{\beta^\alpha x^{\alpha - 1} e^{-\beta x }}{\Gamma(\alpha)}

    Note that this parameterization is equivalent to the above, with
    ``scale = 1 / beta``.


.. _dist gengamma:

* ``gengamma`` **Generalized gamma** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gengamma.html>`_). The probability density function for `gengamma` is:

    .. math::

        f(x, a, c) = \frac{|c| x^{c a-1} \exp(-x^c)}{\Gamma(a)}

    for :math:`x \ge 0`, :math:`a > 0`, and :math:`c \ne 0`.
    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

    `gengamma` takes :math:`a` and :math:`c` as shape parameters.


.. _dist genhalflogistic:

* ``genhalflogistic`` **Generalized Half Logistic** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.genhalflogistic.html>`_). The probability density function for `genhalflogistic` is:

    .. math::

        f(x, c) = \frac{2 (1 - c x)^{1/(c-1)}}{[1 + (1 - c x)^{1/c}]^2}

    for :math:`0 \le x \le 1/c`, and :math:`c > 0`.

    `genhalflogistic` takes ``c`` as a shape parameter for :math:`c`.


.. _dist genhyperbolic:

* ``genhyperbolic`` **Generalized Hyperbolic** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.genhyperbolic.html>`_). The probability density function for `genhyperbolic` is:

    .. math::

        f(x, p, a, b) =
            \frac{(a^2 - b^2)^{p/2}}
            {\sqrt{2\pi}a^{p-0.5}
            K_p\Big(\sqrt{a^2 - b^2}\Big)}
            e^{bx} \times \frac{K_{p - 1/2}
            (a \sqrt{1 + x^2})}
            {(\sqrt{1 + x^2})^{1/2 - p}}

    for :math:`x, p \in ( - \infty; \infty)`,
    :math:`|b| < a` if :math:`p \ge 0`,
    :math:`|b| \le a` if :math:`p < 0`.
    :math:`K_{p}(.)` denotes the modified Bessel function of the second
    kind and order :math:`p` (`scipy.special.kn`)

    `genhyperbolic` takes ``p`` as a tail parameter,
    ``a`` as a shape parameter,
    ``b`` as a skewness parameter.


.. _dist geninvgauss:

* ``geninvgauss`` **Generalized Inverse Gaussian** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.geninvgauss.html>`_). The probability density function for `geninvgauss` is:

    .. math::

        f(x, p, b) = x^{p-1} \exp(-b (x + 1/x) / 2) / (2 K_p(b))

    where `x > 0`, and the parameters `p, b` satisfy `b > 0`.
    :math:`K_p` is the modified Bessel function of second kind of order `p`
    (`scipy.special.kv`).


.. _dist gilbrat:

* ``gilbrat`` **Gilbrat** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gilbrat.html>`_). The probability density function for `gilbrat` is:

    .. math::

        f(x) = \frac{1}{x \sqrt{2\pi}} \exp(-\frac{1}{2} (\log(x))^2)

    `gilbrat` is a special case of `lognorm` with ``s=1``.


.. _dist gompertz:

* ``gompertz`` **Gompertz (Truncated Gumbel)** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gompertz.html>`_). The probability density function for `gompertz` is:

    .. math::

        f(x, c) = c \exp(x) \exp(-c (e^x-1))

    for :math:`x \ge 0`, :math:`c > 0`.

    `gompertz` takes ``c`` as a shape parameter for :math:`c`.


.. _dist gumbel_r:

* ``gumbel_r`` (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gumbel_r.html>`_). The probability density function for `gumbel_r` is:

    .. math::

        f(x) = \exp(-(x + e^{-x}))

    The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
    distribution.  It is also related to the extreme value distribution,
    log-Weibull and Gompertz distributions.


.. _dist gumbel_l:

* ``gumbel_l`` (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gumbel_l.html>`_). The probability density function for `gumbel_l` is:

    .. math::

        f(x) = \exp(x - e^x)

    The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
    distribution.  It is also related to the extreme value distribution,
    log-Weibull and Gompertz distributions.


.. _dist halfcauchy:

* ``halfcauchy`` **Half Cauchy** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.halfcauchy.html>`_). The probability density function for `halfcauchy` is:

    .. math::

        f(x) = \frac{2}{\pi (1 + x^2)}

    for :math:`x \ge 0`.


.. _dist halflogistic:

* ``halflogistic`` **Half Logistic** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.halflogistic.html>`_). The probability density function for `halflogistic` is:

    .. math::

        f(x) = \frac{ 2 e^{-x} }{ (1+e^{-x})^2 }
             = \frac{1}{2} \text{sech}(x/2)^2

    for :math:`x \ge 0`.


.. _dist halfnorm:

* ``halfnorm`` **Half Normal** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.halfnorm.html>`_). The probability density function for `halfnorm` is:

    .. math::

        f(x) = \sqrt{2/\pi} \exp(-x^2 / 2)

    for :math:`x >= 0`.

    `halfnorm` is a special case of `chi` with ``df=1``.


.. _dist halfgennorm:

* ``halfgennorm`` **Generalized Half Normal** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.halfgennorm.html>`_). The probability density function for `halfgennorm` is:

    .. math::

        f(x, \beta) = \frac{\beta}{\Gamma(1/\beta)} \exp(-|x|^\beta)

    for :math:`x > 0`. :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    `gennorm` takes ``beta`` as a shape parameter for :math:`\beta`.
    For :math:`\beta = 1`, it is identical to an exponential distribution.
    For :math:`\beta = 2`, it is identical to a half normal distribution
    (with ``scale=1/sqrt(2)``).


.. _dist hypsecant:

* ``hypsecant`` **Hyperbolic Secant** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.hypsecant.html>`_). The probability density function for `hypsecant` is:

    .. math::

        f(x) = \frac{1}{\pi} \text{sech}(x)

    for a real number :math:`x`.


.. _dist invgamma:

* ``invgamma`` **Inverse Gamma** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.invgamma.html>`_). The probability density function for `invgamma` is:

    .. math::

        f(x, a) = \frac{x^{-a-1}}{\Gamma(a)} \exp(-\frac{1}{x})

    for :math:`x >= 0`, :math:`a > 0`. :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    `invgamma` takes ``a`` as a shape parameter for :math:`a`.

    `invgamma` is a special case of `gengamma` with ``c=-1``, and it is a
    different parameterization of the scaled inverse chi-squared distribution.
    Specifically, if the scaled inverse chi-squared distribution is
    parameterized with degrees of freedom :math:`\nu` and scaling parameter
    :math:`\tau^2`, then it can be modeled using `invgamma` with
    ``a=`` :math:`\nu/2` and ``scale=`` :math:`\nu \tau^2/2`.


.. _dist invgauss:

* ``invgauss`` **Inverse Gaussian** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.invgauss.html>`_). The probability density function for `invgauss` is:

    .. math::

        f(x, \mu) = \frac{1}{\sqrt{2 \pi x^3}}
                    \exp(-\frac{(x-\mu)^2}{2 x \mu^2})

    for :math:`x >= 0` and :math:`\mu > 0`.

    `invgauss` takes ``mu`` as a shape parameter for :math:`\mu`.


.. _dist invweibull:

* ``invweibull`` **Inverse Weibull** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.invweibull.html>`_). The probability density function for `invweibull` is:

    .. math::

        f(x, c) = c x^{-c-1} \exp(-x^{-c})

    for :math:`x > 0`, :math:`c > 0`.

    `invweibull` takes ``c`` as a shape parameter for :math:`c`.


.. _dist johnsonsb:

* ``johnsonsb`` **Johnson SB** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.johnsonsb.html>`_). The probability density function for `johnsonsb` is:

    .. math::

        f(x, a, b) = \frac{b}{x(1-x)}  \phi(a + b \log \frac{x}{1-x} )

    where :math:`x`, :math:`a`, and :math:`b` are real scalars; :math:`b > 0`
    and :math:`x \in [0,1]`.  :math:`\phi` is the pdf of the normal
    distribution.

    `johnsonsb` takes :math:`a` and :math:`b` as shape parameters.


.. _dist johnsonsu:

* ``johnsonsu`` **Johnson SU** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.johnsonsu.html>`_). The probability density function for `johnsonsu` is:

    .. math::

        f(x, a, b) = \frac{b}{\sqrt{x^2 + 1}}
                     \phi(a + b \log(x + \sqrt{x^2 + 1}))

    where :math:`x`, :math:`a`, and :math:`b` are real scalars; :math:`b > 0`.
    :math:`\phi` is the pdf of the normal distribution.

    `johnsonsu` takes :math:`a` and :math:`b` as shape parameters.


.. _dist kappa4:

* ``kappa4`` **Kappa 4 parameter** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kappa4.html>`_). The probability density function for kappa4 is:

    .. math::

        f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1}

    if :math:`h` and :math:`k` are not equal to 0.

    If :math:`h` or :math:`k` are zero then the pdf can be simplified:

    h = 0 and k != 0::

        kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
                              exp(-(1.0 - k*x)**(1.0/k))

    h != 0 and k = 0::

        kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0)

    h = 0 and k = 0::

        kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x))

    kappa4 takes :math:`h` and :math:`k` as shape parameters.

    The kappa4 distribution returns other distributions when certain
    :math:`h` and :math:`k` values are used.

    +------+-------------+----------------+------------------+
    | h    | k=0.0       | k=1.0          | -inf<=k<=inf     |
    +======+=============+================+==================+
    | -1.0 | Logistic    |                | Generalized      |
    |      |             |                | Logistic(1)      |
    |      |             |                |                  |
    |      | logistic(x) |                |                  |
    +------+-------------+----------------+------------------+
    |  0.0 | Gumbel      | Reverse        | Generalized      |
    |      |             | Exponential(2) | Extreme Value    |
    |      |             |                |                  |
    |      | gumbel_r(x) |                | genextreme(x, k) |
    +------+-------------+----------------+------------------+
    |  1.0 | Exponential | Uniform        | Generalized      |
    |      |             |                | Pareto           |
    |      |             |                |                  |
    |      | expon(x)    | uniform(x)     | genpareto(x, -k) |
    +------+-------------+----------------+------------------+


.. _dist kappa3:

* ``kappa3`` **Kappa 3 parameter** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kappa3.html>`_). The probability density function for `kappa3` is:

    .. math::

        f(x, a) = a (a + x^a)^{-(a + 1)/a}

    for :math:`x > 0` and :math:`a > 0`.

    `kappa3` takes ``a`` as a shape parameter for :math:`a`.


.. _dist ksone:

* ``ksone`` **Distribution of Kolmogorov-Smirnov one-sided test statistic** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ksone.html>`_). :math:`D_n^+` and :math:`D_n^-` are given by

    .. math::

        D_n^+ &= \text{sup}_x (F_n(x) - F(x)),\\
        D_n^- &= \text{sup}_x (F(x) - F_n(x)),\\

    where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF.
    `ksone` describes the distribution under the null hypothesis of the KS test
    that the empirical CDF corresponds to :math:`n` i.i.d. random variates
    with CDF :math:`F`.


.. _dist kstwo:

* ``kstwo`` **Distribution of Kolmogorov-Smirnov two-sided test statistic** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kstwo.html>`_). :math:`D_n` is given by

    .. math::

        D_n = \text{sup}_x |F_n(x) - F(x)|

    where :math:`F` is a (continuous) CDF and :math:`F_n` is an empirical CDF.
    `kstwo` describes the distribution under the null hypothesis of the KS test
    that the empirical CDF corresponds to :math:`n` i.i.d. random variates
    with CDF :math:`F`.


.. _dist kstwobign:

* ``kstwobign`` **Limiting Distribution of scaled Kolmogorov-Smirnov two-sided test statistic.** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kstwobign.html>`_). :math:`\sqrt{n} D_n` is given by

    .. math::

        D_n = \text{sup}_x |F_n(x) - F(x)|

    where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF.
    `kstwobign`  describes the asymptotic distribution (i.e. the limit of
    :math:`\sqrt{n} D_n`) under the null hypothesis of the KS test that the
    empirical CDF corresponds to i.i.d. random variates with CDF :math:`F`.


.. _dist laplace:

* ``laplace`` **Laplace** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.laplace.html>`_). The probability density function for `laplace` is

    .. math::

        f(x) = \frac{1}{2} \exp(-|x|)

    for a real number :math:`x`.


.. _dist laplace_asymmetric:

* ``laplace_asymmetric`` (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.laplace_asymmetric.html>`_). The probability density function for `laplace_asymmetric` is

    .. math::

       f(x, \kappa) &= \frac{1}{\kappa+\kappa^{-1}}\exp(-x\kappa),\quad x\ge0\\
                    &= \frac{1}{\kappa+\kappa^{-1}}\exp(x/\kappa),\quad x<0\\

    for :math:`-\infty < x < \infty`, :math:`\kappa > 0`.

    `laplace_asymmetric` takes ``kappa`` as a shape parameter for
    :math:`\kappa`. For :math:`\kappa = 1`, it is identical to a
    Laplace distribution.


.. _dist levy:

* ``levy`` **Levy** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.levy.html>`_). The probability density function for `levy` is:

    .. math::

        f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp\left(-\frac{1}{2x}\right)

    for :math:`x >= 0`.

    This is the same as the Levy-stable distribution with :math:`a=1/2` and
    :math:`b=1`.


.. _dist logistic:

* ``logistic`` **Logistic** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.logistic.html>`_). The probability density function for `logistic` is:

    .. math::

        f(x) = \frac{\exp(-x)}
                    {(1+\exp(-x))^2}

    `logistic` is a special case of `genlogistic` with ``c=1``.

    Remark that the survival function (``logistic.sf``) is equal to the
    Fermi-Dirac distribution describing fermionic statistics.


.. _dist loggamma:

* ``loggamma`` **Log-Gamma** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.loggamma.html>`_). The probability density function for `loggamma` is:

    .. math::

        f(x, c) = \frac{\exp(c x - \exp(x))}
                       {\Gamma(c)}

    for all :math:`x, c > 0`. Here, :math:`\Gamma` is the
    gamma function (`scipy.special.gamma`).

    `loggamma` takes ``c`` as a shape parameter for :math:`c`.


.. _dist loglaplace:

* ``loglaplace`` **Log-Laplace (Log Double Exponential)** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.loglaplace.html>`_). The probability density function for `loglaplace` is:

    .. math::

        f(x, c) = \begin{cases}\frac{c}{2} x^{ c-1}  &\text{for } 0 < x < 1\\
                               \frac{c}{2} x^{-c-1}  &\text{for } x \ge 1
                  \end{cases}

    for :math:`c > 0`.

    `loglaplace` takes ``c`` as a shape parameter for :math:`c`.


.. _dist lognorm:

* ``lognorm`` **Log-Normal** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lognorm.html>`_). The probability density function for `lognorm` is:

    .. math::

        f(x, s) = \frac{1}{s x \sqrt{2\pi}}
                  \exp\left(-\frac{\log^2(x)}{2s^2}\right)

    for :math:`x > 0`, :math:`s > 0`.

    `lognorm` takes ``s`` as a shape parameter for :math:`s`.


.. _dist loguniform:

* ``loguniform`` **Log-Uniform** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.loguniform.html>`_). The probability density function for this class is:

    .. math::

        f(x, a, b) = \frac{1}{x \log(b/a)}

    for :math:`a \le x \le b`, :math:`b > a > 0`. This class takes
    :math:`a` and :math:`b` as shape parameters.


.. _dist lomax:

* ``lomax`` **Lomax (Pareto of the second kind)** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lomax.html>`_). The probability density function for `lomax` is:

    .. math::

        f(x, c) = \frac{c}{(1+x)^{c+1}}

    for :math:`x \ge 0`, :math:`c > 0`.

    `lomax` takes ``c`` as a shape parameter for :math:`c`.

    `lomax` is a special case of `pareto` with ``loc=-1.0``.


.. _dist maxwell:

* ``maxwell`` **Maxwell** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.maxwell.html>`_). A special case of a `chi` distribution,  with ``df=3``, ``loc=0.0``, and given ``scale = a``, where ``a`` is the parameter used in the Mathworld description.

    The probability density function for `maxwell` is:

    .. math::

        f(x) = \sqrt{2/\pi}x^2 \exp(-x^2/2)

    for :math:`x >= 0`.


.. _dist mielke:

* ``mielke`` **Mielke's Beta-Kappa** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.mielke.html>`_). The probability density function for `mielke` is:

    .. math::

        f(x, k, s) = \frac{k x^{k-1}}{(1+x^s)^{1+k/s}}

    for :math:`x > 0` and :math:`k, s > 0`. The distribution is sometimes
    called Dagum distribution. It was already defined in, called
    a Burr Type III distribution (`burr` with parameters ``c=s`` and
    ``d=k/s``).

    `mielke` takes ``k`` and ``s`` as shape parameters.


.. _dist moyal:

* ``moyal`` **Moyal** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.moyal.html>`_). The probability density function for `moyal` is:

    .. math::

        f(x) = \exp(-(x + \exp(-x))/2) / \sqrt{2\pi}

    for a real number :math:`x`.


.. _dist nakagami:

* ``nakagami`` **Nakagami** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.nakagami.html>`_). The probability density function for `nakagami` is:

    .. math::

        f(x, \nu) = \frac{2 \nu^\nu}{\Gamma(\nu)} x^{2\nu-1} \exp(-\nu x^2)

    for :math:`x >= 0`, :math:`\nu > 0`.

    `nakagami` takes ``nu`` as a shape parameter for :math:`\nu`.


.. _dist ncx2:

* ``ncx2`` **Non-central chi-squared** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ncx2.html>`_). The probability density function for `ncx2` is:

    .. math::

        f(x, k, \lambda) = \frac{1}{2} \exp(-(\lambda+x)/2)
            (x/\lambda)^{(k-2)/4}  I_{(k-2)/2}(\sqrt{\lambda x})

    for :math:`x >= 0` and :math:`k, \lambda > 0`. :math:`k` specifies the
    degrees of freedom (denoted ``df`` in the implementation) and
    :math:`\lambda` is the non-centrality parameter (denoted ``nc`` in the
    implementation). :math:`I_\nu` denotes the modified Bessel function of
    first order of degree :math:`\nu` (`scipy.special.iv`).

    `ncx2` takes ``df`` and ``nc`` as shape parameters.


.. _dist ncf:

* ``ncf`` **Non-central F** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ncf.html>`_). The probability density function for `ncf` is:

    .. math::

        f(x, n_1, n_2, \lambda) =
            \exp\left(\frac{\lambda}{2} +
                      \lambda n_1 \frac{x}{2(n_1 x + n_2)}
                \right)
            n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\
            (n_2 + n_1 x)^{-(n_1 + n_2)/2}
            \gamma(n_1/2) \gamma(1 + n_2/2) \\
            \frac{L^{\frac{n_1}{2}-1}_{n_2/2}
                \left(-\lambda n_1 \frac{x}{2(n_1 x + n_2)}\right)}
            {B(n_1/2, n_2/2)
                \gamma\left(\frac{n_1 + n_2}{2}\right)}

    for :math:`n_1, n_2 > 0`, :math:`\lambda \ge 0`.  Here :math:`n_1` is the
    degrees of freedom in the numerator, :math:`n_2` the degrees of freedom in
    the denominator, :math:`\lambda` the non-centrality parameter,
    :math:`\gamma` is the logarithm of the Gamma function, :math:`L_n^k` is a
    generalized Laguerre polynomial and :math:`B` is the beta function.

    `ncf` takes ``df1``, ``df2`` and ``nc`` as shape parameters. If ``nc=0``,
    the distribution becomes equivalent to the Fisher distribution.


.. _dist nct:

* ``nct`` **Non-central Student's T** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.nct.html>`_). If :math:`Y` is a standard normal random variable and :math:`V` is an independent chi-square random variable (`chi2`) with :math:`k` degrees of freedom, then

    .. math::

        X = \frac{Y + c}{\sqrt{V/k}}

    has a non-central Student's t distribution on the real line.
    The degrees of freedom parameter :math:`k` (denoted ``df`` in the
    implementation) satisfies :math:`k > 0` and the noncentrality parameter
    :math:`c` (denoted ``nc`` in the implementation) is a real number.


.. _dist norm:

* ``norm`` **Normal (Gaussian)** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html>`_). The probability density function for `norm` is:

    .. math::

        f(x) = \frac{\exp(-x^2/2)}{\sqrt{2\pi}}

    for a real number :math:`x`.


.. _dist norminvgauss:

* ``norminvgauss`` **Normal Inverse Gaussian** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norminvgauss.html>`_). The probability density function for `norminvgauss` is:

    .. math::

        f(x, a, b) = \frac{a \, K_1(a \sqrt{1 + x^2})}{\pi \sqrt{1 + x^2}} \,
                     \exp(\sqrt{a^2 - b^2} + b x)

    where :math:`x` is a real number, the parameter :math:`a` is the tail
    heaviness and :math:`b` is the asymmetry parameter satisfying
    :math:`a > 0` and :math:`|b| <= a`.
    :math:`K_1` is the modified Bessel function of second kind
    (`scipy.special.k1`).


.. _dist pareto:

* ``pareto`` **Pareto** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.pareto.html>`_). The probability density function for `pareto` is:

    .. math::

        f(x, b) = \frac{b}{x^{b+1}}

    for :math:`x \ge 1`, :math:`b > 0`.

    `pareto` takes ``b`` as a shape parameter for :math:`b`.


.. _dist pearson3:

* ``pearson3`` **Pearson type III** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.pearson3.html>`_). The probability density function for `pearson3` is:

    .. math::

        f(x, \kappa) = \frac{|\beta|}{\Gamma(\alpha)}
                       (\beta (x - \zeta))^{\alpha - 1}
                       \exp(-\beta (x - \zeta))

    where:

    .. math::

            \beta = \frac{2}{\kappa}

            \alpha = \beta^2 = \frac{4}{\kappa^2}

            \zeta = -\frac{\alpha}{\beta} = -\beta

    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).
    Pass the skew :math:`\kappa` into `pearson3` as the shape parameter
    ``skew``.


.. _dist powerlaw:

* ``powerlaw`` **Power-function** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.powerlaw.html>`_). The probability density function for `powerlaw` is:

    .. math::

        f(x, a) = a x^{a-1}

    for :math:`0 \le x \le 1`, :math:`a > 0`.

    `powerlaw` takes ``a`` as a shape parameter for :math:`a`.


.. _dist powerlognorm:

* ``powerlognorm`` **Power log normal** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.powerlognorm.html>`_). The probability density function for `powerlognorm` is:

    .. math::

        f(x, c, s) = \frac{c}{x s} \phi(\log(x)/s)
                     (\Phi(-\log(x)/s))^{c-1}

    where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf,
    and :math:`x > 0`, :math:`s, c > 0`.

    `powerlognorm` takes :math:`c` and :math:`s` as shape parameters.


.. _dist powernorm:

* ``powernorm`` **Power normal** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.powernorm.html>`_). The probability density function for `powernorm` is:

    .. math::

        f(x, c) = c \phi(x) (\Phi(-x))^{c-1}

    where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf,
    and :math:`x >= 0`, :math:`c > 0`.

    `powernorm` takes ``c`` as a shape parameter for :math:`c`.


.. _dist rdist:

* ``rdist`` **R-distribution** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rdist.html>`_). The probability density function for `rdist` is:

    .. math::

        f(x, c) = \frac{(1-x^2)^{c/2-1}}{B(1/2, c/2)}

    for :math:`-1 \le x \le 1`, :math:`c > 0`. `rdist` is also called the
    symmetric beta distribution: if B has a `beta` distribution with
    parameters (c/2, c/2), then X = 2*B - 1 follows a R-distribution with
    parameter c.

    `rdist` takes ``c`` as a shape parameter for :math:`c`.

    This distribution includes the following distribution kernels as
    special cases::

        c = 2:  uniform
        c = 3:  `semicircular`
        c = 4:  Epanechnikov (parabolic)
        c = 6:  quartic (biweight)
        c = 8:  triweight


.. _dist rayleigh:

* ``rayleigh`` **Rayleigh** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rayleigh.html>`_). The probability density function for `rayleigh` is:

    .. math::

        f(x) = x \exp(-x^2/2)

    for :math:`x \ge 0`.

    `rayleigh` is a special case of `chi` with ``df=2``.


.. _dist rice:

* ``rice`` **Rice** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rice.html>`_). The probability density function for `rice` is:

    .. math::

        f(x, b) = x \exp(- \frac{x^2 + b^2}{2}) I_0(x b)

    for :math:`x >= 0`, :math:`b > 0`. :math:`I_0` is the modified Bessel
    function of order zero (`scipy.special.i0`).

    `rice` takes ``b`` as a shape parameter for :math:`b`.


.. _dist recipinvgauss:

* ``recipinvgauss`` **Reciprocal Inverse Gaussian** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.recipinvgauss.html>`_). The probability density function for `recipinvgauss` is:

    .. math::

        f(x, \mu) = \frac{1}{\sqrt{2\pi x}}
                    \exp\left(\frac{-(1-\mu x)^2}{2\mu^2x}\right)

    for :math:`x \ge 0`.

    `recipinvgauss` takes ``mu`` as a shape parameter for :math:`\mu`.


.. _dist semicircular:

* ``semicircular`` **Semicircular** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.semicircular.html>`_). The probability density function for `semicircular` is:

    .. math::

        f(x) = \frac{2}{\pi} \sqrt{1-x^2}

    for :math:`-1 \le x \le 1`.

    The distribution is a special case of `rdist` with `c = 3`.


.. _dist skewcauchy:

* ``skewcauchy`` **Skew Cauchy** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.skewcauchy.html>`_). The probability density function for `skewcauchy` is:

    .. math::

        f(x) = \frac{1}{\pi \left(\frac{x^2}{\left(a\, \text{sign}(x) + 1
                                                   \right)^2} + 1 \right)}

    for a real number :math:`x` and skewness parameter :math:`-1 < a < 1`.

    When :math:`a=0`, the distribution reduces to the usual Cauchy
    distribution.


.. _dist skewnorm:

* ``skewnorm`` **Skew normal** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.skewnorm.html>`_). The pdf is::

        skewnorm.pdf(x, a) = 2 * norm.pdf(x) * norm.cdf(a*x)

  `skewnorm` takes a real number :math:`a` as a skewness parameter.
  When ``a = 0`` the distribution is identical to a normal distribution
  (`norm`).


.. _dist studentized_range:

* ``studentized_range`` (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.studentized_range.html>`_). The probability density function for `studentized_range` is:

    .. math::

         f(x; k, \nu) = \frac{k(k-1)\nu^{\nu/2}}{\Gamma(\nu/2)
                        2^{\nu/2-1}} \int_{0}^{\infty} \int_{-\infty}^{\infty}
                        s^{\nu} e^{-\nu s^2/2} \phi(z) \phi(sx + z)
                        [\Phi(sx + z) - \Phi(z)]^{k-2} \,dz \,ds

    for :math:`x ≥ 0`, :math:`k > 1`, and :math:`\nu > 0`.

    `studentized_range` takes ``k`` for :math:`k` and ``df`` for :math:`\nu`
    as shape parameters.

    When :math:`\nu` exceeds 100,000, an asymptotic approximation (infinite
    degrees of freedom) is used to compute the cumulative distribution
    function.


.. _dist t:

* ``t`` **Student's T** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.t.html>`_). The probability density function for `t` is:

    .. math::

        f(x, \nu) = \frac{\Gamma((\nu+1)/2)}
                        {\sqrt{\pi \nu} \Gamma(\nu/2)}
                    (1+x^2/\nu)^{-(\nu+1)/2}

    where :math:`x` is a real number and the degrees of freedom parameter
    :math:`\nu` (denoted ``df`` in the implementation) satisfies
    :math:`\nu > 0`. :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).


.. _dist trapezoid:

* ``trapezoid`` **Trapezoidal** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.trapezoid.html>`_). The trapezoidal distribution can be represented with an up-sloping line from ``loc`` to ``(loc + c*scale)``, then constant to ``(loc + d*scale)`` and then downsloping from ``(loc + d*scale)`` to ``(loc+scale)``.  This defines the trapezoid base from ``loc`` to ``(loc+scale)`` and the flat top from ``c`` to ``d`` proportional to the position along the base with ``0 <= c <= d <= 1``.  When ``c=d``, this is equivalent to `triang` with the same values for `loc`, `scale` and `c`.

  `trapezoid` takes :math:`c` and :math:`d` as shape parameters.


.. _dist triang:

* ``triang`` **Triangular** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.triang.html>`_). The triangular distribution can be represented with an up-sloping line from ``loc`` to ``(loc + c*scale)`` and then downsloping for ``(loc + c*scale)`` to ``(loc + scale)``.

  `triang` takes ``c`` as a shape parameter for :math:`c`.


.. _dist truncexpon:

* ``truncexpon`` **Truncated Exponential** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.truncexpon.html>`_). The probability density function for `truncexpon` is:

    .. math::

        f(x, b) = \frac{\exp(-x)}{1 - \exp(-b)}

    for :math:`0 <= x <= b`.

    `truncexpon` takes ``b`` as a shape parameter for :math:`b`.


.. _dist truncnorm:

* ``truncnorm`` **Truncated Normal** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.truncnorm.html>`_). The standard form of this distribution is a standard normal truncated to the range [a, b] --- notice that a and b are defined over the domain of the standard normal.  To convert clip values for a specific mean and standard deviation, use::

        a, b = (myclip_a - my_mean) / my_std, (myclip_b - my_mean) / my_std

  `truncnorm` takes :math:`a` and :math:`b` as shape parameters.


.. _dist tukeylambda:

* ``tukeylambda`` **Tukey-Lambda** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.tukeylambda.html>`_). A flexible distribution, able to represent and interpolate between the following distributions:

    - Cauchy                (:math:`lambda = -1`)
    - logistic              (:math:`lambda = 0`)
    - approx Normal         (:math:`lambda = 0.14`)
    - uniform from -1 to 1  (:math:`lambda = 1`)

    `tukeylambda` takes a real number :math:`lambda` (denoted ``lam``
    in the implementation) as a shape parameter.


.. _dist uniform:

* ``uniform`` **Uniform** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.uniform.html>`_). a uniform continuous random variable


.. _dist vonmises:

* ``vonmises`` **Von-Mises (Circular)** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.vonmises.html>`_). The probability density function for `vonmises` and `vonmises_line` is:

    .. math::

        f(x, \kappa) = \frac{ \exp(\kappa \cos(x)) }{ 2 \pi I_0(\kappa) }

    for :math:`-\pi \le x \le \pi`, :math:`\kappa > 0`. :math:`I_0` is the
    modified Bessel function of order zero (`scipy.special.i0`).

    `vonmises` is a circular distribution which does not restrict the
    distribution to a fixed interval. Currently, there is no circular
    distribution framework in scipy. The ``cdf`` is implemented such that
    ``cdf(x + 2*np.pi) == cdf(x) + 1``.

    `vonmises_line` is the same distribution, defined on :math:`[-\pi, \pi]`
    on the real line. This is a regular (i.e. non-circular) distribution.

    `vonmises` and `vonmises_line` take ``kappa`` as a shape parameter.


.. _dist vonmises_line:

* ``vonmises_line`` (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.vonmises_line.html>`_). The probability density function for `vonmises` and `vonmises_line` is:

    .. math::

        f(x, \kappa) = \frac{ \exp(\kappa \cos(x)) }{ 2 \pi I_0(\kappa) }

    for :math:`-\pi \le x \le \pi`, :math:`\kappa > 0`. :math:`I_0` is the
    modified Bessel function of order zero (`scipy.special.i0`).

    `vonmises` is a circular distribution which does not restrict the
    distribution to a fixed interval. Currently, there is no circular
    distribution framework in scipy. The ``cdf`` is implemented such that
    ``cdf(x + 2*np.pi) == cdf(x) + 1``.

    `vonmises_line` is the same distribution, defined on :math:`[-\pi, \pi]`
    on the real line. This is a regular (i.e. non-circular) distribution.

    `vonmises` and `vonmises_line` take ``kappa`` as a shape parameter.


.. _dist wald:

* ``wald`` **Wald** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.wald.html>`_). The probability density function for `wald` is:

    .. math::

        f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp(- \frac{ (x-1)^2 }{ 2x })

    for :math:`x >= 0`.

    `wald` is a special case of `invgauss` with ``mu=1``.


.. _dist weibull_min:

* ``weibull_min`` (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.weibull_min.html>`_). The probability density function for `weibull_min` is:

    .. math::

        f(x, c) = c x^{c-1} \exp(-x^c)

    for :math:`x > 0`, :math:`c > 0`.

    `weibull_min` takes ``c`` as a shape parameter for :math:`c`.
    (named :math:`k` in Wikipedia article and :math:`a` in
    ``numpy.random.weibull``).  Special shape values are :math:`c=1` and
    :math:`c=2` where Weibull distribution reduces to the `expon` and
    `rayleigh` distributions respectively.


.. _dist weibull_max:

* ``weibull_max`` (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.weibull_max.html>`_). The probability density function for `weibull_max` is:

    .. math::

        f(x, c) = c (-x)^{c-1} \exp(-(-x)^c)

    for :math:`x < 0`, :math:`c > 0`.

    `weibull_max` takes ``c`` as a shape parameter for :math:`c`.


.. _dist wrapcauchy:

* ``wrapcauchy`` **Wrapped Cauchy** (`help <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.wrapcauchy.html>`_). The probability density function for `wrapcauchy` is:

    .. math::

        f(x, c) = \frac{1-c^2}{2\pi (1+c^2 - 2c \cos(x))}

    for :math:`0 \le x \le 2\pi`, :math:`0 < c < 1`.

    `wrapcauchy` takes ``c`` as a shape parameter for :math:`c`.


.. code to create: see blog/agg/examples/probems_and_solutions.ipynb