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# scipy.stats.lognorm

scipy.stats.lognorm = <scipy.stats._continuous_distns.lognorm_gen object>

  A lognormal continuous random variable.

  As an instance of the `rv_continuous` class, `lognorm` 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.

  Notes

  The probability density function for `lognorm` is:

  \[f(x, s) = \frac{1}{s x \sqrt{2\pi}} \exp\left(-\frac{\log^2(x)}{2s^2}\right)\]

  for \(x > 0\), \(s > 0\).

  `lognorm` takes `s` as a shape parameter for \(s\).

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `lognorm.pdf(x, s, loc, scale)` is identically equivalent to `lognorm.pdf(y, s) / scale` with `y = (x - loc) / scale`. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.

  Suppose a normally distributed random variable `X` has mean `mu` and standard deviation `sigma`. Then `Y = exp(X)` is lognormally distributed with `s = sigma` and `scale = exp(mu)`.

  Examples

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

  Calculate the first four moments:

      >>> s = 0.954
      >>> mean, var, skew, kurt = lognorm.stats(s, moments='mvsk')

  Display the probability density function (`pdf`):

      >>> x = np.linspace(lognorm.ppf(0.01, s),
      ...                 lognorm.ppf(0.99, s), 100)
      >>> ax.plot(x, lognorm.pdf(x, s),
      ...        'r-', lw=5, alpha=0.6, label='lognorm pdf')

  Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

  Freeze the distribution and display the frozen `pdf`:

      >>> rv = lognorm(s)
      >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

  Check accuracy of `cdf` and `ppf`:

      >>> vals = lognorm.ppf([0.001, 0.5, 0.999], s)
      >>> np.allclose([0.001, 0.5, 0.999], lognorm.cdf(vals, s))
      True

  Generate random numbers:

      >>> r = lognorm.rvs(s, size=1000)

  And compare the histogram:

      >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
      >>> ax.set_xlim([x[0], x[-1]])
      >>> ax.legend(loc='best', frameon=False)
      >>> plt.show()

  The logarithm of a log-normally distributed random variable is normally distributed:

      >>> import numpy as np
      >>> import matplotlib.pyplot as plt
      >>> from scipy import stats
      >>> fig, ax = plt.subplots(1, 1)
      >>> mu, sigma = 2, 0.5
      >>> X = stats.norm(loc=mu, scale=sigma)
      >>> Y = stats.lognorm(s=sigma, scale=np.exp(mu))
      >>> x = np.linspace(*X.interval(0.999))
      >>> y = Y.rvs(size=10000)
      >>> ax.plot(x, X.pdf(x), label='X (pdf)')
      >>> ax.hist(np.log(y), density=True, bins=x, label='log(Y) (histogram)')
      >>> ax.legend()
      >>> plt.show()

  Methods

  rvs(s, loc=0, scale=1, size=1, random_state=None) | Random variates.
  pdf(x, s, loc=0, scale=1) | Probability density function.
  logpdf(x, s, loc=0, scale=1) | Log of the probability density function.
  cdf(x, s, loc=0, scale=1) | Cumulative distribution function.
  logcdf(x, s, loc=0, scale=1) | Log of the cumulative distribution function.
  sf(x, s, loc=0, scale=1) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(x, s, loc=0, scale=1) | Log of the survival function.
  ppf(q, s, loc=0, scale=1) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, s, loc=0, scale=1) | Inverse survival function (inverse of `sf`).
  moment(order, s, loc=0, scale=1) | Non-central moment of the specified order.
  stats(s, loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(s, loc=0, scale=1) | (Differential) entropy of the RV.
  fit(data) | Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
  expect(func, args=(s,), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) | Expected value of a function (of one argument) with respect to the distribution.
  median(s, loc=0, scale=1) | Median of the distribution.
  mean(s, loc=0, scale=1) | Mean of the distribution.
  var(s, loc=0, scale=1) | Variance of the distribution.
  std(s, loc=0, scale=1) | Standard deviation of the distribution.
  interval(confidence, s, loc=0, scale=1) | Confidence interval with equal areas around the median.