scipy.stats.lognorm — SciPy v1.15.2 Manual (original) (raw)

scipy.stats.lognorm = <scipy.stats._continuous_distns.lognorm_gen object>[source]#

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 withy = (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