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In SciPy, use scipy.stats.expon for exponential waiting times. Its scale parameter is the mean waiting time, not the rate: if your model uses rate lambda, set scale=1/lambda. Then use cdf for the probability of an event occurring by a time, sf for the probability it occurs after that time, and rvs to generate values.
What SciPy’s exponential distribution models
SciPy describes scipy.stats.expon as “An exponential continuous random variable.” In its standard form, the distribution has density exp(-x) for x >= 0. It is commonly used to model nonnegative waiting times when the model assumes a constant event rate.
The standard distribution starts at zero. SciPy represents a shifted and scaled value using y = (x - loc) / scale; the default parameters are loc=0 and scale=1. The location shifts the support origin, while scale stretches or compresses the distribution. This location shift is not a separate “noncentral” exponential model. See the SciPy 1.16.0 expon API reference and the SciPy statistics tutorial.
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Convert a rate to SciPy’s scale
Many probability texts parameterize an exponential distribution by rate lambda. SciPy’s expon uses scale instead, so enter the reciprocal: scale = 1 / lambda. In the zero-location model, scale is also the mean waiting time.
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For example, a rate of 0.2 events per time unit corresponds to a mean wait of 5 time units. Set scale=5, not scale=0.2. Keep the units consistent: a rate per minute produces a scale in minutes.
from scipy.stats import expon
rate = 0.2 # events per time unit
rv = expon(scale=1 / rate) # mean waiting time: 5 time unitsPass parameters by keyword to make the intended meaning clear. For a zero-origin exponential with mean 3, use expon(scale=3). The official SciPy statistics tutorial recommends explicit loc and scale keywords.
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Calculate probabilities and generate samples
Import expon from scipy.stats. You can call its methods directly or freeze parameters in a distribution object, as in rv below. With rate 0.2 per time unit, the following computes the chance of an event within five time units, the chance of waiting longer than five, and a reproducible array of 1,000 generated waiting times.
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rate = 0.2
rv = expon(scale=1 / rate)
prob_within_5 = rv.cdf(5)
prob_after_5 = rv.sf(5)
samples = rv.rvs(size=1000, random_state=42)Use cdf(x) for the probability that a value is at or below x, and sf(x) for the probability it is above x. For an upper-tail probability, prefer sf(x) to writing 1 - cdf(x): SciPy notes that the survival function can be more accurate in some cases.
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Choose the method that matches the question
| Question | Method | What it returns |
|---|---|---|
| What is the density at a value? | pdf(x) |
Probability density, not the probability of one exact continuous value |
| What is the log density? | logpdf(x) |
Natural logarithm of the probability density |
| What is the probability of a value at or below a threshold? | cdf(x) |
Cumulative probability |
| What is the probability of a value above a threshold? | sf(x) |
Survival, or upper-tail, probability |
| What is the log cumulative or log upper-tail probability? | logcdf(x) or logsf(x) |
Logarithm of the corresponding tail probability |
| What value marks a cumulative probability? | ppf(q) |
Quantile for cumulative probability q |
| What value marks an upper-tail probability? | isf(q) |
Inverse survival quantile for upper-tail probability q |
| How can I draw random values? | rvs(size=..., random_state=...) |
Random variates |
| How can I inspect moments or support? | stats(...) or support() |
Distribution statistics or support bounds |
For example, rv.ppf(0.95) gives the 95th percentile, while rv.isf(0.05) gives the threshold with 5% of the distribution above it. These are equivalent quantile questions expressed from opposite tails.
Use a shifted origin only when the model requires it
Set loc when the distribution should begin at a value other than zero. For instance, expon(loc=10, scale=3) shifts the support to start at 10 while retaining scale 3. A location shift changes the origin; it does not change scale into a rate or produce a different named distribution. The default support begins at loc.
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Distinguish related SciPy distributions
expon is the exponential distribution and is a special case of the gamma distribution with shape a=1. Do not confuse it with exponnorm, SciPy’s exponentially modified normal distribution, which is a different model. The definitions and methods for expon are in the versioned SciPy API reference.
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