scipy.stats.boltzmann = <scipy.stats._discrete_distns.boltzmann_gen object at 0x2b45d304a710>[source]

A Boltzmann (Truncated Discrete Exponential) random variable.

Discrete 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

lambda_, N : array_like

shape parameters

loc : array_like, optional

location parameter (default=0)

size : int or tuple of ints, optional

shape of random variates (default computed from input arguments )

moments : str, 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’)

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

location parameters returning a “frozen” discrete RV object:

rv = boltzmann(lambda_, N, loc=0)

  • Frozen RV object with the same methods but holding the given shape and location fixed.


The probability mass function for boltzmann is:

boltzmann.pmf(k) = (1-exp(-lambda_)*exp(-lambda_*k)/(1-exp(-lambda_*N))

for k = 0,..., N-1.

boltzmann takes lambda_ and N as shape parameters.


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

Calculate a few first 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, 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)

(Source code)


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 variates.
pmf(x, lambda_, N, loc=0) Probability mass function.
logpmf(x, lambda_, N, loc=0) Log of the probability mass function.
cdf(x, lambda_, N, loc=0) Cumulative density function.
logcdf(x, lambda_, N, loc=0) Log of the cumulative density function.
sf(x, lambda_, N, loc=0) Survival function (1-cdf — sometimes more accurate).
logsf(x, 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, 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(alpha, lambda_, N, loc=0) Endpoints of the range that contains alpha percent of the distribution

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