scipy.stats.boltzmann = <scipy.stats.distributions.boltzmann_gen object at 0x4ddfe10>[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:

Parameters :

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
>>> [ lambda_, N ] = [<Replace with reasonable values>]
>>> rv = boltzmann(lambda_, N)

Display frozen pmf

>>> x = np.arange(0, np.minimum(rv.dist.b, 3))
>>> h = plt.vlines(x, 0, rv.pmf(x), lw=2)

Here, rv.dist.b is the right endpoint of the support of rv.dist.

Check accuracy of cdf and ppf

>>> prb = boltzmann.cdf(x, lambda_, N)
>>> h = plt.semilogy(np.abs(x - boltzmann.ppf(prb, lambda_, N)) + 1e-20)

Random number generation

>>> R = boltzmann.rvs(lambda_, N, size=100)


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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