scipy.stats.planck

scipy.stats.planck = <scipy.stats._discrete_distns.planck_gen object>

A Planck discrete exponential random variable.

As an instance of the rv_discrete class, planck 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.

Methods

rvs(lambda_, loc=0, size=1, random_state=None)	Random variates.
pmf(k, lambda_, loc=0)	Probability mass function.
logpmf(k, lambda_, loc=0)	Log of the probability mass function.
cdf(k, lambda_, loc=0)	Cumulative distribution function.
logcdf(k, lambda_, loc=0)	Log of the cumulative distribution function.
sf(k, lambda_, loc=0)	Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(k, lambda_, loc=0)	Log of the survival function.
ppf(q, lambda_, loc=0)	Percent point function (inverse of cdf — percentiles).
isf(q, lambda_, loc=0)	Inverse survival function (inverse of sf).
stats(lambda_, loc=0, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(lambda_, loc=0)	(Differential) entropy of the RV.
expect(func, args=(lambda_,), loc=0, lb=None, ub=None, conditional=False)	Expected value of a function (of one argument) with respect to the distribution.
median(lambda_, loc=0)	Median of the distribution.
mean(lambda_, loc=0)	Mean of the distribution.
var(lambda_, loc=0)	Variance of the distribution.
std(lambda_, loc=0)	Standard deviation of the distribution.
interval(confidence, lambda_, loc=0)	Confidence interval with equal areas around the median.

See also

geom

Notes

The probability mass function for planck is:

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

for \(k \ge 0\) and \(\lambda > 0\).

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

The probability mass function above is defined in the “standardized” form. To shift distribution use the loc parameter. Specifically, planck.pmf(k, lambda_, loc) is identically equivalent to planck.pmf(k - loc, lambda_).

Examples

Try it in your browser!

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

Calculate the first four moments:

>>> lambda_ = 0.51
>>> mean, var, skew, kurt = planck.stats(lambda_, moments='mvsk')

Display the probability mass function (pmf):

>>> x = np.arange(planck.ppf(0.01, lambda_),
...               planck.ppf(0.99, lambda_))
>>> ax.plot(x, planck.pmf(x, lambda_), 'bo', ms=8, label='planck pmf')
>>> ax.vlines(x, 0, planck.pmf(x, lambda_), colors='b', lw=5, alpha=0.5)

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

Freeze the distribution and display the frozen pmf:

>>> rv = planck(lambda_)
>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
...         label='frozen pmf')
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

Check accuracy of cdf and ppf:

>>> prob = planck.cdf(x, lambda_)
>>> np.allclose(x, planck.ppf(prob, lambda_))
True

Generate random numbers:

>>> r = planck.rvs(lambda_, size=1000)

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