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[Skip to main content](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#main-content) Back to top [  SciPy](https://docs.scipy.org/doc/scipy/index.html) - [Installing](https://scipy.org/install/) - [User Guide](https://docs.scipy.org/doc/scipy/tutorial/index.html) - [API reference](https://docs.scipy.org/doc/scipy/reference/index.html) - [Building from source](https://docs.scipy.org/doc/scipy/building/index.html) - [Development](https://docs.scipy.org/doc/scipy/dev/index.html) - [Release notes](https://docs.scipy.org/doc/scipy/release.html) Choose version - [GitHub](https://github.com/scipy/scipy "GitHub") - [Scientific Python Forum](https://discuss.scientific-python.org/c/contributor/scipy/ "Scientific Python Forum") - [Installing](https://scipy.org/install/) - [User Guide](https://docs.scipy.org/doc/scipy/tutorial/index.html) - [API reference](https://docs.scipy.org/doc/scipy/reference/index.html) - [Building from source](https://docs.scipy.org/doc/scipy/building/index.html) - [Development](https://docs.scipy.org/doc/scipy/dev/index.html) - [Release notes](https://docs.scipy.org/doc/scipy/release.html) Choose version - [GitHub](https://github.com/scipy/scipy "GitHub") - [Scientific Python Forum](https://discuss.scientific-python.org/c/contributor/scipy/ "Scientific Python Forum") Search `Ctrl`\+`K` Section Navigation - [scipy](https://docs.scipy.org/doc/scipy/reference/main_namespace.html) - [scipy.cluster](https://docs.scipy.org/doc/scipy/reference/cluster.html) - [scipy.constants](https://docs.scipy.org/doc/scipy/reference/constants.html) - [scipy.datasets](https://docs.scipy.org/doc/scipy/reference/datasets.html) - [scipy.differentiate](https://docs.scipy.org/doc/scipy/reference/differentiate.html) - [scipy.fft](https://docs.scipy.org/doc/scipy/reference/fft.html) - [scipy.fftpack](https://docs.scipy.org/doc/scipy/reference/fftpack.html) - [scipy.integrate](https://docs.scipy.org/doc/scipy/reference/integrate.html) - [scipy.interpolate](https://docs.scipy.org/doc/scipy/reference/interpolate.html) - [scipy.io](https://docs.scipy.org/doc/scipy/reference/io.html) - [scipy.linalg](https://docs.scipy.org/doc/scipy/reference/linalg.html) - [scipy.ndimage](https://docs.scipy.org/doc/scipy/reference/ndimage.html) - [scipy.odr](https://docs.scipy.org/doc/scipy/reference/odr.html) - [scipy.optimize](https://docs.scipy.org/doc/scipy/reference/optimize.html) - [scipy.signal](https://docs.scipy.org/doc/scipy/reference/signal.html) - [scipy.sparse](https://docs.scipy.org/doc/scipy/reference/sparse.html) - [scipy.spatial](https://docs.scipy.org/doc/scipy/reference/spatial.html) - [scipy.special](https://docs.scipy.org/doc/scipy/reference/special.html) - [scipy.stats](https://docs.scipy.org/doc/scipy/reference/stats.html) - [SciPy API](https://docs.scipy.org/doc/scipy/reference/index.html) - [Statistical functions (`scipy.stats`)](https://docs.scipy.org/doc/scipy/reference/stats.html) - scipy.stats.poisson # scipy.stats.poisson[\#](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#scipy-stats-poisson "Link to this heading") scipy.stats.poisson *\= \<scipy.stats.\_discrete\_distns.poisson\_gen object\>*[\[source\]](https://github.com/scipy/scipy/blob/v1.17.0/scipy/stats/_discrete_distns.py#L967-L1030)[\#](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#scipy.stats.poisson "Link to this definition") A Poisson discrete random variable. As an instance of the [`rv_discrete`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_discrete.html#scipy.stats.rv_discrete "scipy.stats.rv_discrete") class, [`poisson`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#scipy.stats.poisson "scipy.stats.poisson") object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution. Methods | | | |---|---| | **rvs(mu, loc=0, size=1, random\_state=None)** | Random variates. | | **pmf(k, mu, loc=0)** | Probability mass function. | | **logpmf(k, mu, loc=0)** | Log of the probability mass function. | | **cdf(k, mu, loc=0)** | Cumulative distribution function. | | **logcdf(k, mu, loc=0)** | Log of the cumulative distribution function. | | **sf(k, mu, loc=0)** | Survival function (also defined as `1 - cdf`, but *sf* is sometimes more accurate). | | **logsf(k, mu, loc=0)** | Log of the survival function. | | **ppf(q, mu, loc=0)** | Percent point function (inverse of `cdf` — percentiles). | | **isf(q, mu, loc=0)** | Inverse survival function (inverse of `sf`). | | **stats(mu, loc=0, moments=’mv’)** | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). | | **entropy(mu, loc=0)** | (Differential) entropy of the RV. | | **expect(func, args=(mu,), loc=0, lb=None, ub=None, conditional=False)** | Expected value of a function (of one argument) with respect to the distribution. | | **median(mu, loc=0)** | Median of the distribution. | | **mean(mu, loc=0)** | Mean of the distribution. | | **var(mu, loc=0)** | Variance of the distribution. | | **std(mu, loc=0)** | Standard deviation of the distribution. | | **interval(confidence, mu, loc=0)** | Confidence interval with equal areas around the median. | Notes The probability mass function for [`poisson`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#scipy.stats.poisson "scipy.stats.poisson") is: \\\[f(k) = \\exp(-\\mu) \\frac{\\mu^k}{k!}\\\] for \\(k \\ge 0\\). [`poisson`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#scipy.stats.poisson "scipy.stats.poisson") takes \\(\\mu \\geq 0\\) as shape parameter. When \\(\\mu = 0\\), the `pmf` method returns `1.0` at quantile \\(k = 0\\). The probability mass function above is defined in the “standardized” form. To shift distribution use the `loc` parameter. Specifically, `poisson.pmf(k, mu, loc)` is identically equivalent to `poisson.pmf(k - loc, mu)`. Examples Try it in your browser\! ``` >>> import numpy as np >>> from scipy.stats import poisson >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) ``` Get the support: ``` >>> mu = 0.6 >>> lb, ub = poisson.support(mu) ``` Calculate the first four moments: ``` >>> mean, var, skew, kurt = poisson.stats(mu, moments='mvsk') ``` Display the probability mass function (`pmf`): ``` >>> x = np.arange(poisson.ppf(0.01, mu), ... poisson.ppf(0.99, mu)) >>> ax.plot(x, poisson.pmf(x, mu), 'bo', ms=8, label='poisson pmf') >>> ax.vlines(x, 0, poisson.pmf(x, mu), colors='b', lw=5, alpha=0.5) ``` Alternatively, the distribution object can be called (as a function) to fix the shape and location. This returns a “frozen” RV object holding the given parameters fixed. Freeze the distribution and display the frozen `pmf`: ``` >>> rv = poisson(mu) >>> 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 = poisson.cdf(x, mu) >>> np.allclose(x, poisson.ppf(prob, mu)) True ``` Generate random numbers: ``` >>> r = poisson.rvs(mu, size=1000) ``` Go Back Open In Tab [previous scipy.stats.planck](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.planck.html "previous page") [next scipy.stats.poisson\_binom](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson_binom.html "next page") On this page - [`poisson`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#scipy.stats.poisson) © Copyright 2008, The SciPy community. Created using [Sphinx](https://www.sphinx-doc.org/) 8.1.3. 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scipy.stats.poisson *\= \<scipy.stats.\_discrete\_distns.poisson\_gen object\>*[\[source\]](https://github.com/scipy/scipy/blob/v1.17.0/scipy/stats/_discrete_distns.py#L967-L1030)[\#](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#scipy.stats.poisson "Link to this definition") A Poisson discrete random variable. As an instance of the [`rv_discrete`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_discrete.html#scipy.stats.rv_discrete "scipy.stats.rv_discrete") class, [`poisson`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#scipy.stats.poisson "scipy.stats.poisson") object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution. Methods | | | |---|---| | **rvs(mu, loc=0, size=1, random\_state=None)** | Random variates. | | **pmf(k, mu, loc=0)** | Probability mass function. | | **logpmf(k, mu, loc=0)** | Log of the probability mass function. | | **cdf(k, mu, loc=0)** | Cumulative distribution function. | | **logcdf(k, mu, loc=0)** | Log of the cumulative distribution function. | | **sf(k, mu, loc=0)** | Survival function (also defined as `1 - cdf`, but *sf* is sometimes more accurate). | | **logsf(k, mu, loc=0)** | Log of the survival function. | | **ppf(q, mu, loc=0)** | Percent point function (inverse of `cdf` — percentiles). | | **isf(q, mu, loc=0)** | Inverse survival function (inverse of `sf`). | | **stats(mu, loc=0, moments=’mv’)** | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). | | **entropy(mu, loc=0)** | (Differential) entropy of the RV. | | **expect(func, args=(mu,), loc=0, lb=None, ub=None, conditional=False)** | Expected value of a function (of one argument) with respect to the distribution. | | **median(mu, loc=0)** | Median of the distribution. | | **mean(mu, loc=0)** | Mean of the distribution. | | **var(mu, loc=0)** | Variance of the distribution. | | **std(mu, loc=0)** | Standard deviation of the distribution. | | **interval(confidence, mu, loc=0)** | Confidence interval with equal areas around the median. | Notes The probability mass function for [`poisson`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#scipy.stats.poisson "scipy.stats.poisson") is: \\\[f(k) = \\exp(-\\mu) \\frac{\\mu^k}{k!}\\\] for \\(k \\ge 0\\). [`poisson`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#scipy.stats.poisson "scipy.stats.poisson") takes \\(\\mu \\geq 0\\) as shape parameter. When \\(\\mu = 0\\), the `pmf` method returns `1.0` at quantile \\(k = 0\\). The probability mass function above is defined in the “standardized” form. To shift distribution use the `loc` parameter. Specifically, `poisson.pmf(k, mu, loc)` is identically equivalent to `poisson.pmf(k - loc, mu)`. Examples ``` >>> import numpy as np >>> from scipy.stats import poisson >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) ``` Get the support: ``` >>> mu = 0.6 >>> lb, ub = poisson.support(mu) ``` Calculate the first four moments: ``` >>> mean, var, skew, kurt = poisson.stats(mu, moments='mvsk') ``` Display the probability mass function (`pmf`): ``` >>> x = np.arange(poisson.ppf(0.01, mu), ... poisson.ppf(0.99, mu)) >>> ax.plot(x, poisson.pmf(x, mu), 'bo', ms=8, label='poisson pmf') >>> ax.vlines(x, 0, poisson.pmf(x, mu), colors='b', lw=5, alpha=0.5) ``` Alternatively, the distribution object can be called (as a function) to fix the shape and location. This returns a “frozen” RV object holding the given parameters fixed. Freeze the distribution and display the frozen `pmf`: ``` >>> rv = poisson(mu) >>> 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 = poisson.cdf(x, mu) >>> np.allclose(x, poisson.ppf(prob, mu)) True ``` Generate random numbers: ``` >>> r = poisson.rvs(mu, size=1000) ```