scipy.stats.crystalball = <scipy.stats._continuous_distns.crystalball_gen object>[source]

Crystalball distribution

As an instance of the rv_continuous class, crystalball object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.


The probability density function for crystalball is:

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

where \(A = (m / |beta|)**n * exp(-beta**2 / 2)\), \(B = m/|beta| - |beta|\) and \(N\) is a normalisation constant.

crystalball takes \(\beta\) and \(m\) as shape parameters. \(\beta\) defines the point where the pdf changes from a power-law to a gaussian distribution \(m\) is power of the power-law tail.


[1]“Crystal Ball Function”,

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, crystalball.pdf(x, beta, m, loc, scale) is identically equivalent to crystalball.pdf(y, beta, m) / scale with y = (x - loc) / scale.

New in version 0.19.0.


>>> from scipy.stats import crystalball
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate a few first moments:

>>> beta, m = 2, 3
>>> mean, var, skew, kurt = crystalball.stats(beta, m, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(crystalball.ppf(0.01, beta, m),
...                 crystalball.ppf(0.99, beta, m), 100)
>>> ax.plot(x, crystalball.pdf(x, beta, m),
...        'r-', lw=5, alpha=0.6, label='crystalball pdf')

Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

Freeze the distribution and display the frozen pdf:

>>> rv = crystalball(beta, m)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = crystalball.ppf([0.001, 0.5, 0.999], beta, m)
>>> np.allclose([0.001, 0.5, 0.999], crystalball.cdf(vals, beta, m))

Generate random numbers:

>>> r = crystalball.rvs(beta, m, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)


rvs(beta, m, loc=0, scale=1, size=1, random_state=None) Random variates.
pdf(x, beta, m, loc=0, scale=1) Probability density function.
logpdf(x, beta, m, loc=0, scale=1) Log of the probability density function.
cdf(x, beta, m, loc=0, scale=1) Cumulative distribution function.
logcdf(x, beta, m, loc=0, scale=1) Log of the cumulative distribution function.
sf(x, beta, m, loc=0, scale=1) Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(x, beta, m, loc=0, scale=1) Log of the survival function.
ppf(q, beta, m, loc=0, scale=1) Percent point function (inverse of cdf — percentiles).
isf(q, beta, m, loc=0, scale=1) Inverse survival function (inverse of sf).
moment(n, beta, m, loc=0, scale=1) Non-central moment of order n
stats(beta, m, loc=0, scale=1, moments=’mv’) Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(beta, m, loc=0, scale=1) (Differential) entropy of the RV.
fit(data, beta, m, loc=0, scale=1) Parameter estimates for generic data.
expect(func, args=(beta, m), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) Expected value of a function (of one argument) with respect to the distribution.
median(beta, m, loc=0, scale=1) Median of the distribution.
mean(beta, m, loc=0, scale=1) Mean of the distribution.
var(beta, m, loc=0, scale=1) Variance of the distribution.
std(beta, m, loc=0, scale=1) Standard deviation of the distribution.
interval(alpha, beta, m, loc=0, scale=1) Endpoints of the range that contains alpha percent of the distribution

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