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Repair common Windows errors and clear accumulated junk for a smoother, more stable PC - no reinstall needed.Free scan · no reinstallUse scipy.stats.poisson to calculate exact and cumulative count probabilities, upper-tail probabilities, quantiles, and random Poisson samples. Its parameter mu is the expected count for the interval or exposure you are modeling; loc, when used, shifts the distribution’s support rather than changing that expected count.
What the Poisson distribution models
SciPy defines poisson as a discrete Poisson random variable. For a nonnegative integer count k, its probability mass function is exp(-mu) * mu**k / k!, with mu >= 0. The standard support is the integer counts starting at zero. SciPy’s Poisson reference
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Choose mu to match the observation interval or exposure in your model. For example, a rate expressed per hour must be converted consistently if you are modeling a different duration; the API does not select the interval or infer a rate for you. Under the Poisson model, the theoretical mean and variance are both mu, and the standard deviation is sqrt(mu).
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Calculate probabilities and samples
Import the distribution object and pass the count threshold and mu to the method that matches your question:
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from scipy.stats import poisson
mu = 3.0
exactly_two = poisson.pmf(2, mu)
at_most_two = poisson.cdf(2, mu)
more_than_two = poisson.sf(2, mu)
quantile_95 = poisson.ppf(0.95, mu)
samples = poisson.rvs(mu, size=1000, random_state=0)The example shows API usage; it is not a claim of an executed test. Method definitions and parameters are documented in the SciPy Poisson reference.
| Question | Method | Meaning |
|---|---|---|
What is the probability of exactly k events? |
poisson.pmf(k, mu) |
Probability mass at count k. |
What is the probability of at most k events? |
poisson.cdf(k, mu) |
Probability that the count is less than or equal to k. |
What is the probability of more than k events? |
poisson.sf(k, mu) |
Upper-tail probability, equivalent in meaning to 1 - cdf. |
What count corresponds to probability q? |
poisson.ppf(q, mu) |
The discrete quantile at cumulative probability q. |
| How can I generate observations? | poisson.rvs(mu, size=..., random_state=...) |
Random draws from the distribution; size sets the output shape. |
| What are the theoretical moments? | poisson.mean(mu), poisson.var(mu), poisson.std(mu) or poisson.stats(mu) |
Mean, variance, standard deviation, or a moments summary. |
Choose the right tail or quantile calculation
Use sf for an upper tail
poisson.sf(k, mu) means the probability of a count greater than k. It is often the direct choice for questions such as “more than five events.” SciPy notes that sf can be more accurate than calculating 1 - poisson.cdf(k, mu), especially when the CDF is close to one. SciPy Poisson API
Read ppf as a discrete quantile
For a discrete distribution, the CDF advances in steps. poisson.ppf(q, mu) returns the smallest integer count whose CDF is at least q; it is not a continuous inverse that can return any real-valued count. SciPy probability distributions tutorial
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Understand mu, loc, and zero
mu sets the Poisson count scale
mu is the nonnegative shape parameter in the probability formula and represents the expected count over the chosen interval or exposure. It is not a per-unit rate unless the chosen interval is that unit.
loc shifts the support
With loc, SciPy shifts the support: poisson.pmf(k, mu, loc) is equivalent to poisson.pmf(k - loc, mu). This shifts which values of the random variable are supported; it does not replace or modify mu. SciPy Poisson API
mu = 0 is a documented edge case
SciPy documents that when mu = 0, the PMF returns 1.0 at k = 0. SciPy Poisson API
Use the discrete API conventions
Poisson counts are discrete, so use pmf rather than the continuous-distribution method pdf. SciPy’s discrete-distribution conventions also do not provide a scale parameter or estimation methods such as fit. Check the SciPy version installed in your environment against the documentation for that version if relying on version-specific behavior. SciPy probability distributions tutorial
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