scipy.stats.truncnorm

scipy.stats.truncnorm = <scipy.stats._continuous_distns.truncnorm_gen object>

A truncated normal continuous random variable.

As an instance of the rv_continuous class, truncnorm 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

This distribution is the normal distribution centered on loc (default 0), with standard deviation scale (default 1), and truncated at a and b standard deviations from loc. For arbitrary loc and scale, a and b are not the abscissae at which the shifted and scaled distribution is truncated.

Note

If a_trunc and b_trunc are the abscissae at which we wish to truncate the distribution (as opposed to the number of standard deviations from loc), then we can calculate the distribution parameters a and b as follows:

a, b = (a_trunc - loc) / scale, (b_trunc - loc) / scale

This is a common point of confusion. For additional clarification, please see the example below.

Examples

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>>> import numpy as np
>>> from scipy.stats import truncnorm
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> a, b = 0.1, 2
>>> mean, var, skew, kurt = truncnorm.stats(a, b, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(truncnorm.ppf(0.01, a, b),
...                 truncnorm.ppf(0.99, a, b), 100)
>>> ax.plot(x, truncnorm.pdf(x, a, b),
...        'r-', lw=5, alpha=0.6, label='truncnorm 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 = truncnorm(a, b)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = truncnorm.ppf([0.001, 0.5, 0.999], a, b)
>>> np.allclose([0.001, 0.5, 0.999], truncnorm.cdf(vals, a, b))
True

Generate random numbers:

>>> r = truncnorm.rvs(a, b, 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()

In the examples above, loc=0 and scale=1, so the plot is truncated at a on the left and b on the right. However, suppose we were to produce the same histogram with loc = 1 and scale=0.5.

>>> loc, scale = 1, 0.5
>>> rv = truncnorm(a, b, loc=loc, scale=scale)
>>> x = np.linspace(truncnorm.ppf(0.01, a, b),
...                 truncnorm.ppf(0.99, a, b), 100)
>>> r = rv.rvs(size=1000)

>>> fig, ax = plt.subplots(1, 1)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
>>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim(a, b)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

Note that the distribution is no longer appears to be truncated at abscissae a and b. That is because the standard normal distribution is first truncated at a and b, then the resulting distribution is scaled by scale and shifted by loc. If we instead want the shifted and scaled distribution to be truncated at a and b, we need to transform these values before passing them as the distribution parameters.

>>> a_transformed, b_transformed = (a - loc) / scale, (b - loc) / scale
>>> rv = truncnorm(a_transformed, b_transformed, loc=loc, scale=scale)
>>> x = np.linspace(truncnorm.ppf(0.01, a, b),
...                 truncnorm.ppf(0.99, a, b), 100)
>>> r = rv.rvs(size=10000)

>>> fig, ax = plt.subplots(1, 1)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
>>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim(a-0.1, b+0.1)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

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Methods

rvs(a, b, loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, a, b, loc=0, scale=1)	Probability density function.
logpdf(x, a, b, loc=0, scale=1)	Log of the probability density function.
cdf(x, a, b, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, a, b, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, a, b, loc=0, scale=1)	Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(x, a, b, loc=0, scale=1)	Log of the survival function.
ppf(q, a, b, loc=0, scale=1)	Percent point function (inverse of cdf — percentiles).
isf(q, a, b, loc=0, scale=1)	Inverse survival function (inverse of sf).
moment(order, a, b, loc=0, scale=1)	Non-central moment of the specified order.
stats(a, b, loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(a, b, 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=(a, b), 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(a, b, loc=0, scale=1)	Median of the distribution.
mean(a, b, loc=0, scale=1)	Mean of the distribution.
var(a, b, loc=0, scale=1)	Variance of the distribution.
std(a, b, loc=0, scale=1)	Standard deviation of the distribution.
interval(confidence, a, b, loc=0, scale=1)	Confidence interval with equal areas around the median.