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# 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.

  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

      >>> 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)

  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.