# numpy.random.normal

random.normal(loc=0.0, scale=1.0, size=None)

  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].

  Note

  New code should use the `normal` method of a `default_rng()` instance instead; please see the Quick Start.

  Parameters
    loc float or array_like of floats

      Mean (“centre”) of the distribution.

    scale float or array_like of floats

      Standard deviation (spread or “width”) of the distribution. Must be non-negative.

    size int 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
    out ndarray or scalar

      Drawn samples from the parameterized normal distribution.

  See also

  `scipy.stats.norm`

    probability density function, distribution or cumulative density function, etc.

  `Generator.normal`

    which should be used for new code.

  Notes

  The probability density for the Gaussian distribution 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 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.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.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

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