scipy.stats.lognorm#
- 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
takess
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
andscale
parameters. Specifically,lognorm.pdf(x, s, loc, scale)
is identically equivalent tolognorm.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 meanmu
and standard deviationsigma
. ThenY = exp(X)
is lognormally distributed withs = sigma
andscale = exp(mu)
.Examples
>>> 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
andppf
:>>> 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, histtype='stepfilled', alpha=0.2) >>> ax.legend(loc='best', frameon=False) >>> 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.