numpy.random.normal — NumPy v1.18.dev0 Manual (2024)

Draw random samples from a normal (Gaussian) distribution.

The probability density function of the normal distribution, firstderived by De Moivre and 200 years later by both Gauss and Laplaceindependently [2], is often called the bell curve because ofits characteristic shape (see the example below).

The normal distributions occurs often in nature. For example, itdescribes the commonly occurring distribution of samples influencedby a large number of tiny, random disturbances, each with its ownunique distribution [2].

Note

New code should use the normal method of a default_rng()instance instead; see random-quick-start.

Parameters

locfloat or array_like of floats: Mean (“centre”) of the distribution.
scalefloat or array_like of floats: Standard deviation (spread or “width”) of the distribution. Must benon-negative.
sizeint or tuple of ints, optional: Output shape. If the given shape is, e.g., (m, n, k), thenm * n * k samples are drawn. If size is None (default),a single value is returned if loc and scale are both scalars.Otherwise, np.broadcast(loc, scale).size samples are drawn.

Returns

outndarray or scalar: Drawn samples from the parameterized normal distribution.

Notes

The probability density for the Gaussian distribution is

System Message: WARNING/2 (p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }}e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },)

latex exited with error[stdout]This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=latex) restricted \write18 enabled.entering extended mode(./math.texLaTeX2e <2017-04-15>Babel <3.18> and hyphenation patterns for 3 language(s) loaded.(/usr/share/texlive/texmf-dist/tex/latex/base/article.clsDocument Class: article 2014/09/29 v1.4h Standard LaTeX document class(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty(/usr/share/texlive/texmf-dist/tex/latex/base/utf8.def(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.dfu)(/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu)(/usr/share/texlive/texmf-dist/tex/latex/base/omsenc.dfu)))(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.styFor additional information on amsmath, use the `?' option.(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))! LaTeX Error: File `anyfontsize.sty' not found.Type X to quit or <RETURN> to proceed,or enter new name. (Default extension: sty)Enter file name: ! Emergency stop.<read *> l.8 \usepackage {bm}^^MNo pages of output.Transcript written on math.log.

where

System Message: WARNING/2 (\mu)

is the mean and

System Message: WARNING/2 (\sigma)

the standarddeviation. The square of the standard deviation,

System Message: WARNING/2 (\sigma^2)

,is called the variance.

The function has its peak at the mean, and its “spread” increases withthe standard deviation (the function reaches 0.607 times its maximum at

System Message: WARNING/2 (x + \sigma)

and

System Message: WARNING/2 (x - \sigma)

[2]). This implies thatnormal is more likely to return samples lying close to the mean, ratherthan those far away.

References

1: Wikipedia, “Normal distribution”,https://en.wikipedia.org/wiki/Normal_distribution
2(1,2,3): P. R. Peebles Jr., “Central Limit Theorem” in “Probability,Random Variables and Random Signal Principles”, 4th ed., 2001,pp. 51, 51, 125.

Examples

Draw samples from the distribution:

>>> mu, sigma = 0, 0.1 # mean and standard deviation>>> s = np.random.normal(mu, sigma, 1000)

Verify the mean and the variance:

>>> abs(mu - np.mean(s))0.0 # may vary

>>> abs(sigma - np.std(s, ddof=1))0.1 # may vary

Display the histogram of the samples, along withthe probability density function:

>>> import matplotlib.pyplot as plt>>> count, bins, ignored = plt.hist(s, 30, density=True)>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *...  np.exp( - (bins - mu)**2 / (2 * sigma**2) ),...  linewidth=2, color='r')>>> plt.show()

Two-by-four array of samples from N(3, 6.25):

>>> np.random.normal(3, 2.5, size=(2, 4))array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random