Draw random samples from a normal (Gaussian) distribution.
The probability density function of the normal distribution, first
derived by De Moivre and 200 years later by both Gauss and Laplace
independently [2], is often called the bell curve because of
its characteristic shape (see the example below).
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].
- 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 be
non-negative.
- sizeint or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k)
, then
m * 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.
See also
scipy.stats.norm
probability density function, distribution or cumulative density function, etc.
Notes
The probability density for the Gaussian distribution is
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where
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is the mean and
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the standard
deviation. The square of the standard deviation,
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,
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
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and
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[2]). This implies that
normal
is more likely to return samples lying close to the
mean, rather than 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.default_rng().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 with
the 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.default_rng().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