# numpy.random.wald

random.wald(mean, scale, size=None)

  Draw samples from a Wald, or inverse Gaussian, distribution.

  As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal.

  The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time.

  Note

  New code should use the `wald` method of a `Generator` instance instead; please see the Quick Start.

  Parameters:
    mean float or array_like of floats

      Distribution mean, must be > 0.

    scale float or array_like of floats

      Scale parameter, must be > 0.

    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 `mean` and `scale` are both scalars. Otherwise, `np.broadcast(mean, scale).size` samples are drawn.

  Returns:
    out ndarray or scalar

      Drawn samples from the parameterized Wald distribution.

  See also

  `random.Generator.wald`

    which should be used for new code.

  Notes

  The probability density function for the Wald distribution is

  \[P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^ \frac{-scale(x-mean)^2}{2\cdotp mean^2x}\]

  As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes.

  References

  [1]

  Brighton Webs Ltd., Wald Distribution, https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp

  [2]

  Chhikara, Raj S., and Folks, J. Leroy, “The Inverse Gaussian Distribution: Theory : Methodology, and Applications”, CRC Press, 1988.

  [3]

  Wikipedia, “Inverse Gaussian distribution” https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

  Examples

  Draw values from the distribution and plot the histogram:

      >>> import matplotlib.pyplot as plt
      >>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, density=True)
      >>> plt.show()