larray.random.normal

larray.random.normal(loc=0.0, scale=1.0, axes=None, meta=None)

Draw random samples from a normal (Gaussian) distribution.

Its probability density function is often called the bell curve because of its characteristic shape (see the example below)

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.

axesint, tuple of int, str, Axis or tuple/list/AxisCollection of Axis, optional

Minimum axes the resulting array must have. Defaults to None. The resulting array axes will be the union of those mentioned in axes and those of loc and scale. If loc and scale are scalars and axes is None, a single value is returned. Otherwise, if the resulting axes have a shape of, e.g., (m, n, k), then m * n * k samples are drawn.

metalist of pairs or dict or OrderedDict or Metadata, optional

Metadata (title, description, author, creation_date, …) associated with the array. Keys must be strings. Values must be of type string, int, float, date, time or datetime.

Returns

LArray or scalar

Drawn samples from the parameterized normal distribution.

Notes

The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2].

The probability density function for the Gaussian distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2], is

\[p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },\]

where \(\mu\) is the mean and \(\sigma\) the standard deviation. The square of the standard deviation, \(\sigma^2\), is called the variance.

The function has its peak at the mean, and its “spread” increases with the standard deviation (the function reaches 0.607 times its maximum at \(x + \sigma\) and \(x - \sigma\) [2]). This implies that la.random.normal is more likely to return samples lying close to the mean, rather than those far away.

References

1

Wikipedia, “Normal distribution”, http://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

Generate a 2 x 3 array with numbers drawn from the distribution:

>>> la.random.normal(0, 1, axes=(2, 3))                                         # doctest: +SKIP
{0}*\{1}*                   0                     1                   2
        0  0.3564325741877542    0.8944149721039006  1.7206904920773107
        1  0.6904447654719367  -0.09395966570976753   0.185136309092257

With named and labelled axes

>>> la.random.normal(0, 1, axes='a=a0,a1;b=b0..b2')                             # doctest: +SKIP
a\b                  b0                   b1                   b2
 a0  2.3096106652701827  -0.4269082412118316  -1.0862791566867225
 a1  0.8598817639620348   -2.386411240813283  0.10116503197279443

With varying loc and scale (each depending on a different axis)

>>> a = la.Axis('a=a0,a1')
>>> b = la.Axis('b=b0..b2')
>>> mu = la.sequence(a, initial=5, inc=5)
>>> mu
a  a0  a1
    5  10
>>> sigma = la.sequence(b, initial=1)
>>> sigma
b  b0  b1  b2
    1   2   3
>>> la.random.normal(mu, sigma)                                                 # doctest: +SKIP
a\b                  b0                  b1                  b2
 a0   5.939369790854615  2.5043856460438403    8.33560126941519
 a1  10.759526714752091  10.093213549397403  11.705881778249683

Draw 1000 samples from the distribution:

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

Verify the mean and the variance:

>>> abs(mu - la.mean(sample)) < 0.01
True
>>> abs(sigma - la.std(sample, ddof=1)) < 0.01
True

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

>>> import matplotlib.pyplot as plt                                         # doctest: +SKIP
>>> count, bins, ignored = plt.hist(sample, 30, normed=True)                # doctest: +SKIP
>>> pdf = 1 / (sigma * la.sqrt(2 * la.pi)) \
...       * la.exp(- (bins - mu) ** 2 / (2 * sigma ** 2))                   # doctest: +SKIP
>>> _ = plt.plot(bins, pdf, linewidth=2, color='r')                         # doctest: +SKIP
>>> plt.show()                                                              # doctest: +SKIP