numpy.random.
lognormal
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 default_rng() instance instead; see random-quick-start.
default_rng()
Mean value of the underlying normal distribution. Default is 0.
Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1.
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.
(m, n, k)
m * n * k
None
mean
sigma
np.broadcast(mean, sigma).size
Drawn samples from the parameterized log-normal distribution.
See also
scipy.stats.lognorm
probability density function, distribution, cumulative density function, etc.
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:
System Message: WARNING/2 (p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})} )
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where
System Message: WARNING/2 (\mu)
System Message: WARNING/2 (\sigma)
References
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
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.product(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))
>>> plt.plot(x, pdf, color='r', linewidth=2) >>> plt.show()