A Skellam discrete random variable.
Discrete random variables are defined from a standard form and may require some shape parameters to complete its specification. Any optional keyword parameters can be passed to the methods of the RV object as given below:
Parameters: | x : array_like
q : array_like
mu1, mu2 : array_like
loc : array_like, optional
size : int or tuple of ints, optional
moments : str, optional
Alternatively, the object may be called (as a function) to fix the shape and location parameters returning a “frozen” discrete RV object: rv = skellam(mu1, mu2, loc=0)
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Notes
Probability distribution of the difference of two correlated or uncorrelated Poisson random variables.
Let k1 and k2 be two Poisson-distributed r.v. with expected values lam1 and lam2. Then, k1 - k2 follows a Skellam distribution with parameters mu1 = lam1 - rho*sqrt(lam1*lam2) and mu2 = lam2 - rho*sqrt(lam1*lam2), where rho is the correlation coefficient between k1 and k2. If the two Poisson-distributed r.v. are independent then rho = 0.
Parameters mu1 and mu2 must be strictly positive.
For details see: http://en.wikipedia.org/wiki/Skellam_distribution
skellam takes mu1 and mu2 as shape parameters.
Examples
>>> from scipy.stats import skellam
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate a few first moments:
>>> mu1, mu2 = 15, 8
>>> mean, var, skew, kurt = skellam.stats(mu1, mu2, moments='mvsk')
Display the probability mass function (pmf):
>>> x = np.arange(skellam.ppf(0.01, mu1, mu2),
... skellam.ppf(0.99, mu1, mu2))
>>> ax.plot(x, skellam.pmf(x, mu1, mu2), 'bo', ms=8, label='skellam pmf')
>>> ax.vlines(x, 0, skellam.pmf(x, mu1, mu2), colors='b', lw=5, alpha=0.5)
Alternatively, freeze the distribution and display the frozen pmf:
>>> rv = skellam(mu1, mu2)
>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
... label='frozen pmf')
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
Check accuracy of cdf and ppf:
>>> prob = skellam.cdf(x, mu1, mu2)
>>> np.allclose(x, skellam.ppf(prob, mu1, mu2))
True
Generate random numbers:
>>> r = skellam.rvs(mu1, mu2, size=1000)
Methods
rvs(mu1, mu2, loc=0, size=1) | Random variates. |
pmf(x, mu1, mu2, loc=0) | Probability mass function. |
logpmf(x, mu1, mu2, loc=0) | Log of the probability mass function. |
cdf(x, mu1, mu2, loc=0) | Cumulative density function. |
logcdf(x, mu1, mu2, loc=0) | Log of the cumulative density function. |
sf(x, mu1, mu2, loc=0) | Survival function (1-cdf — sometimes more accurate). |
logsf(x, mu1, mu2, loc=0) | Log of the survival function. |
ppf(q, mu1, mu2, loc=0) | Percent point function (inverse of cdf — percentiles). |
isf(q, mu1, mu2, loc=0) | Inverse survival function (inverse of sf). |
stats(mu1, mu2, loc=0, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). |
entropy(mu1, mu2, loc=0) | (Differential) entropy of the RV. |
expect(func, mu1, mu2, loc=0, lb=None, ub=None, conditional=False) | Expected value of a function (of one argument) with respect to the distribution. |
median(mu1, mu2, loc=0) | Median of the distribution. |
mean(mu1, mu2, loc=0) | Mean of the distribution. |
var(mu1, mu2, loc=0) | Variance of the distribution. |
std(mu1, mu2, loc=0) | Standard deviation of the distribution. |
interval(alpha, mu1, mu2, loc=0) | Endpoints of the range that contains alpha percent of the distribution |