scipy.stats.boltzmann = <scipy.stats._discrete_distns.boltzmann_gen object>[source]#

A Boltzmann (Truncated Discrete Exponential) random variable.

As an instance of the rv_discrete class, boltzmann 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 mass function for boltzmann is:

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

for \(k = 0,..., N-1\).

boltzmann takes \(\lambda > 0\) and \(N > 0\) as shape parameters.

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


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

Calculate the first four moments:

>>> lambda_, N = 1.4, 19
>>> mean, var, skew, kurt = boltzmann.stats(lambda_, N, moments='mvsk')

Display the probability mass function (pmf):

>>> x = np.arange(boltzmann.ppf(0.01, lambda_, N),
...               boltzmann.ppf(0.99, lambda_, N))
>>> ax.plot(x, boltzmann.pmf(x, lambda_, N), 'bo', ms=8, label='boltzmann pmf')
>>> ax.vlines(x, 0, boltzmann.pmf(x, lambda_, N), 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 = boltzmann(lambda_, N)
>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
...         label='frozen pmf')
>>> ax.legend(loc='best', frameon=False)

Check accuracy of cdf and ppf:

>>> prob = boltzmann.cdf(x, lambda_, N)
>>> np.allclose(x, boltzmann.ppf(prob, lambda_, N))

Generate random numbers:

>>> r = boltzmann.rvs(lambda_, N, size=1000)


rvs(lambda_, N, loc=0, size=1, random_state=None)

Random variates.

pmf(k, lambda_, N, loc=0)

Probability mass function.

logpmf(k, lambda_, N, loc=0)

Log of the probability mass function.

cdf(k, lambda_, N, loc=0)

Cumulative distribution function.

logcdf(k, lambda_, N, loc=0)

Log of the cumulative distribution function.

sf(k, lambda_, N, loc=0)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(k, lambda_, N, loc=0)

Log of the survival function.

ppf(q, lambda_, N, loc=0)

Percent point function (inverse of cdf — percentiles).

isf(q, lambda_, N, loc=0)

Inverse survival function (inverse of sf).

stats(lambda_, N, loc=0, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(lambda_, N, loc=0)

(Differential) entropy of the RV.

expect(func, args=(lambda_, N), loc=0, lb=None, ub=None, conditional=False)

Expected value of a function (of one argument) with respect to the distribution.

median(lambda_, N, loc=0)

Median of the distribution.

mean(lambda_, N, loc=0)

Mean of the distribution.

var(lambda_, N, loc=0)

Variance of the distribution.

std(lambda_, N, loc=0)

Standard deviation of the distribution.

interval(confidence, lambda_, N, loc=0)

Confidence interval with equal areas around the median.