numpy.random.Generator.poisson

method

random.Generator.poisson(lam=1.0, size=None)

Draw samples from a Poisson distribution.

The Poisson distribution is the limit of the binomial distribution for large N.

Parameters:

lamfloat or array_like of floats

Expected number of events occurring in a fixed-time interval, must be >= 0. A sequence must be broadcastable over the requested size.

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 lam is a scalar. Otherwise, np.array(lam).size samples are drawn.

Returns:

outndarray or scalar

Drawn samples from the parameterized Poisson distribution.

Notes

The probability mass function (PMF) of Poisson distribution is

\[f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}\]

For events with an expected separation \(\lambda\) the Poisson distribution \(f(k; \lambda)\) describes the probability of \(k\) events occurring within the observed interval \(\lambda\).

Because the output is limited to the range of the C int64 type, a ValueError is raised when lam is within 10 sigma of the maximum representable value.

References

[1]

Weisstein, Eric W. “Poisson Distribution.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/PoissonDistribution.html

[2]

Wikipedia, “Poisson distribution”, https://en.wikipedia.org/wiki/Poisson_distribution

Examples

Draw samples from the distribution:

>>> rng = np.random.default_rng()
>>> lam, size = 5, 10000
>>> s = rng.poisson(lam=lam, size=size)

Verify the mean and variance, which should be approximately lam:

>>> s.mean(), s.var()
(4.9917 5.1088311)  # may vary

Display the histogram and probability mass function:

>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> x = np.arange(0, 21)
>>> pmf = stats.poisson.pmf(x, mu=lam)
>>> plt.hist(s, bins=x, density=True, width=0.5)
>>> plt.stem(x, pmf, 'C1-')
>>> plt.show()

Draw each 100 values for lambda 100 and 500:

>>> s = rng.poisson(lam=(100., 500.), size=(100, 2))