A Mielke’s Beta-Kappa continuous random variable.

Continuous random variables are defined from a standard form and may require some shape parameters to complete its specification. Any optional keyword parameters can be passed to the methods of the RV object as given below:


x : array-like


q : array-like

lower or upper tail probability

k,s : array-like

shape parameters

loc : array-like, optional

location parameter (default=0)

scale : array-like, optional

scale parameter (default=1)

size : int or tuple of ints, optional

shape of random variates (default computed from input arguments )

moments : string, optional

composed of letters [‘mvsk’] specifying which moments to compute where ‘m’ = mean, ‘v’ = variance, ‘s’ = (Fisher’s) skew and ‘k’ = (Fisher’s) kurtosis. (default=’mv’)


mielke.rvs(k,s,loc=0,scale=1,size=1) :

  • random variates

mielke.pdf(x,k,s,loc=0,scale=1) :

  • probability density function

mielke.cdf(x,k,s,loc=0,scale=1) :

  • cumulative density function

mielke.sf(x,k,s,loc=0,scale=1) :

  • survival function (1-cdf — sometimes more accurate)

mielke.ppf(q,k,s,loc=0,scale=1) :

  • percent point function (inverse of cdf — percentiles)

mielke.isf(q,k,s,loc=0,scale=1) :

  • inverse survival function (inverse of sf)

mielke.stats(k,s,loc=0,scale=1,moments=’mv’) :

  • mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’)

mielke.entropy(k,s,loc=0,scale=1) :

  • (differential) entropy of the RV.,k,s,loc=0,scale=1) :

  • Parameter estimates for mielke data

Alternatively, the object may be called (as a function) to fix the shape, :

location, and scale parameters returning a “frozen” continuous RV object: :

rv = mielke(k,s,loc=0,scale=1) :

  • frozen RV object with the same methods but holding the given shape, location, and scale fixed


>>> import matplotlib.pyplot as plt
>>> numargs = mielke.numargs
>>> [ k,s ] = [0.9,]*numargs
>>> rv = mielke(k,s)

Display frozen pdf

>>> x = np.linspace(0,np.minimum(rv.dist.b,3))
>>> h=plt.plot(x,rv.pdf(x))

Check accuracy of cdf and ppf

>>> prb = mielke.cdf(x,k,s)
>>> h=plt.semilogy(np.abs(x-mielke.ppf(prb,c))+1e-20)

Random number generation

>>> R = mielke.rvs(k,s,size=100)

Mielke’s Beta-Kappa distribution

mielke.pdf(x,k,s) = k*x**(k-1) / (1+x**s)**(1+k/s) for x > 0.

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