Use scipy.stats.poisson to calculate exact-count probabilities, cumulative probabilities and tail probabilities, find integer quantiles, or generate random counts. Its key parameter, mu, is the expected count for the interval or exposure you are modeling; loc shifts the count support and does not replace mu.
What does scipy.stats.poisson model?
scipy.stats.poisson is SciPy’s discrete Poisson random-variable object. For a nonnegative integer count k, its probability mass function is exp(-mu) * mu**k / k!, where mu must be nonnegative. The standard support is 0, 1, 2, and so on. SciPy’s Poisson reference documents the formula, parameter, support, and methods.
Choose mu for the exposure you mean
mu is both the theoretical mean and variance of the modeled count; the standard deviation is sqrt(mu). Set it for the time interval, area, or other exposure represented by your random variable. SciPy performs the distribution calculations, but it does not choose the modeling window or determine whether a Poisson model fits your data.
Which SciPy method answers your question?
| Question | Method | Meaning |
|---|---|---|
What is the probability of exactly k events? |
pmf(k, mu) |
Probability mass at count k. |
What is the probability of at most k events? |
cdf(k, mu) |
Probability that the count is less than or equal to k. |
What is the probability of more than k events? |
sf(k, mu) |
Upper-tail probability, equivalent to probability greater than k. |
| What count marks a given cumulative probability? | ppf(q, mu) |
The smallest integer count whose CDF is at least q. |
| How can I generate random counts? | rvs(mu, size=...) |
Random draws from the Poisson distribution. |
| What are the distribution summaries? | mean(mu), var(mu), std(mu), stats(mu) |
Mean, variance, standard deviation, or a summary of distribution statistics. |
The survival function sf can be more accurate than calculating an upper tail as 1 - cdf, especially when the CDF is close to one. Use sf(k, mu) for the probability of strictly more than k; it is not the probability of at least k.
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Calculate probabilities, a quantile, and samples
This example shows the method pattern for a Poisson distribution with mu = 3.0. It is an illustrative code example, not a report of an executed test.
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)Here, exactly_two is the probability mass at 2; at_most_two includes counts 0 through 2; and more_than_two covers counts 3 and above. quantile_95 is an integer count, while samples contains 1,000 generated draws. The method names and parameter pattern follow the SciPy Poisson API reference.
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Understand ppf for a discrete distribution
For a continuous distribution, an inverse CDF may be understood as solving for a point on a smoothly increasing curve. A Poisson CDF instead jumps between integer counts. Accordingly, poisson.ppf(q, mu) returns the smallest integer x for which the cumulative probability is at least q; it does not return a fractional count. This stepwise behavior is described in SciPy’s probability distributions tutorial.
Use loc only when you need a shifted support
The standard Poisson support starts at zero. The loc argument shifts that support: SciPy defines poisson.pmf(k, mu, loc) as equivalent to poisson.pmf(k - loc, mu). It changes where the distribution is located, not its Poisson rate or expected count parameter mu. See the API definition of loc.
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Use discrete-distribution methods, not continuous ones
For SciPy’s discrete distributions, calculate point probabilities with pmf, not pdf. The discrete-distributions conventions in the SciPy tutorial also note that these distributions do not use a scale parameter and do not provide estimation methods such as fit. Do not adapt continuous-distribution examples by swapping in a Poisson object while leaving pdf or scale in place.
Handle the zero-rate edge case
When mu is zero, the distribution places all its probability on count zero. SciPy specifically documents that poisson.pmf(0, 0) returns 1.0. The version 1.16.1 API reference states this edge case.
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Check the SciPy version in your environment
The Poisson API details above are documented in SciPy v1.16.1; the general discrete-distribution conventions are documented in the v1.18.0 tutorial. Check your installed SciPy version and its corresponding documentation when relying on version-specific behavior.
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