# scipy.stats.truncnorm#

scipy.stats.truncnorm = <scipy.stats._continuous_distns.truncnorm_gen object>[source]#

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

```>>> 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, 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()
```

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.