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# numpy.random.lognormal

random.lognormal(mean=0.0, sigma=1.0, size=None)

  Draw samples from a log-normal distribution.

  Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from.

  Note

  New code should use the `lognormal` method of a `Generator` instance instead; please see the Quick start.

  Parameters:
    mean float or array_like of floats, optional

      Mean value of the underlying normal distribution. Default is 0.

    sigma float or array_like of floats, optional

      Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1.

    size int or tuple of ints, optional

      Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `mean` and `sigma` are both scalars. Otherwise, `np.broadcast(mean, sigma).size` samples are drawn.

  Returns:
    out ndarray or scalar

      Drawn samples from the parameterized log-normal distribution.

  See also

  `scipy.stats.lognorm`

    probability density function, distribution, cumulative density function, etc.

  `random.Generator.lognormal`

    which should be used for new code.

  Notes

  A variable x has a log-normal distribution if log(x) is normally distributed. The probability density function for the log-normal distribution is:

  \[p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}\]

  where \(\mu\) is the mean and \(\sigma\) is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the product of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the sum of a large number of independent, identically-distributed variables.

  References

  [1]

  Limpert, E., Stahel, W. A., and Abbt, M., “Log-normal Distributions across the Sciences: Keys and Clues,” BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf

  [2]

  Reiss, R.D. and Thomas, M., “Statistical Analysis of Extreme Values,” Basel: Birkhauser Verlag, 2001, pp. 31-32.

  Examples

  Draw samples from the distribution:

      >>> mu, sigma = 3., 1. # mean and standard deviation
      >>> s = np.random.lognormal(mu, sigma, 1000)

  Display the histogram of the samples, along with the probability density function:

      >>> import matplotlib.pyplot as plt
      >>> count, bins, ignored = plt.hist(s, 100, density=True, align='mid')

      >>> x = np.linspace(min(bins), max(bins), 10000)
      >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
      ...        / (x * sigma * np.sqrt(2 * np.pi)))

      >>> plt.plot(x, pdf, linewidth=2, color='r')
      >>> plt.axis('tight')
      >>> plt.show()

  Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function.

      >>> # Generate a thousand samples: each is the product of 100 random
      >>> # values, drawn from a normal distribution.
      >>> b = []
      >>> for i in range(1000):
      ...    a = 10. + np.random.standard_normal(100)
      ...    b.append(np.prod(a))

      >>> b = np.array(b) / np.min(b) # scale values to be positive
      >>> count, bins, ignored = plt.hist(b, 100, density=True, align='mid')
      >>> sigma = np.std(np.log(b))
      >>> mu = np.mean(np.log(b))

      >>> x = np.linspace(min(bins), max(bins), 10000)
      >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
      ...        / (x * sigma * np.sqrt(2 * np.pi)))

      >>> plt.plot(x, pdf, color='r', linewidth=2)
      >>> plt.show()