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# scipy.stats.bernoulli

scipy.stats.bernoulli = <scipy.stats._discrete_distns.bernoulli_gen object>

  A Bernoulli discrete random variable.

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

  Notes

  The probability mass function for `bernoulli` is:

  \[\begin{split}f(k) = \begin{cases}1-p &\text{if } k = 0\\ p &\text{if } k = 1\end{cases}\end{split}\]

  for \(k\) in \(\{0, 1\}\), \(0 \leq p \leq 1\)

  `bernoulli` takes \(p\) as shape parameter, where \(p\) is the probability of a single success and \(1-p\) is the probability of a single failure.

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

  Examples

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

  Get the support:

      >>> p = 0.3
      >>> lb, ub = bernoulli.support(p)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = bernoulli.stats(p, moments='mvsk')

  Display the probability mass function (`pmf`):

      >>> x = np.arange(bernoulli.ppf(0.01, p),
      ...               bernoulli.ppf(0.99, p))
      >>> ax.plot(x, bernoulli.pmf(x, p), 'bo', ms=8, label='bernoulli pmf')
      >>> ax.vlines(x, 0, bernoulli.pmf(x, p), 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 = bernoulli(p)
      >>> 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 = bernoulli.cdf(x, p)
      >>> np.allclose(x, bernoulli.ppf(prob, p))
      True

  Generate random numbers:

      >>> r = bernoulli.rvs(p, size=1000)