numpy.random.lognormal¶

numpy.random.
lognormal
(mean=0.0, sigma=1.0, size=None)¶ Draw samples from a lognormal distribution.
Draw samples from a lognormal 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.
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. Should be greater than zero. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g.,
(m, n, k)
, thenm * n * k
samples are drawn. If size isNone
(default), a single value is returned ifmean
andsigma
are both scalars. Otherwise,np.broadcast(mean, sigma).size
samples are drawn.Returns: out : ndarray or scalar
Drawn samples from the parameterized lognormal distribution.
See also
scipy.stats.lognorm
 probability density function, distribution, cumulative density function, etc.
Notes
A variable x has a lognormal distribution if log(x) is normally distributed. The probability density function for the lognormal distribution is:
where is the mean and is the standard deviation of the normally distributed logarithm of the variable. A lognormal distribution results if a random variable is the product of a large number of independent, identicallydistributed variables in the same way that a normal distribution results if the variable is the sum of a large number of independent, identicallydistributed variables.
References
[R244] Limpert, E., Stahel, W. A., and Abbt, M., “Lognormal Distributions across the Sciences: Keys and Clues,” BioScience, Vol. 51, No. 5, May, 2001. http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf [R245] Reiss, R.D. and Thomas, M., “Statistical Analysis of Extreme Values,” Basel: Birkhauser Verlag, 2001, pp. 3132. 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, normed=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 lognormal 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.random(100) ... b.append(np.product(a))
>>> b = np.array(b) / np.min(b) # scale values to be positive >>> count, bins, ignored = plt.hist(b, 100, normed=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()