Frozen Distributions¶

A frozen distribution in scipy.stats is a distribution object with its parameters fixed (frozen) at creation time. This allows you to call methods like .rvs(), .pdf(), .cdf(), and .ppf() without repeatedly passing the distribution parameters.

Creating a Frozen Distribution¶

When you call a distribution constructor with specific parameters, you get a frozen distribution object:

import scipy.stats as stats

# Create a frozen normal distribution with mean=3.0, std=1.0
a = stats.norm(loc=3.0)

# Generate random samples from the frozen distribution
samples = a.rvs(size=(2, 3), random_state=1)
print(samples)
print(type(samples))   # <class 'numpy.ndarray'>
print(samples.shape)   # (2, 3)
print(samples.dtype)   # float64

The output is always a NumPy array, and the random_state parameter ensures reproducibility.

Frozen vs Unfrozen Usage¶

Without a frozen distribution, you must pass parameters every time:

# Unfrozen — parameters passed each time
stats.norm.pdf(0, loc=3.0, scale=1.0)
stats.norm.cdf(0, loc=3.0, scale=1.0)
stats.norm.rvs(loc=3.0, scale=1.0, size=5)

# Frozen — parameters set once
a = stats.norm(loc=3.0, scale=1.0)
a.pdf(0)
a.cdf(0)
a.rvs(size=5)

Frozen distributions are cleaner, less error-prone, and more efficient when you need to call multiple methods on the same distribution.

Available Methods¶

Every frozen distribution object provides a consistent interface:

Method	Description
`rvs(size, random_state)`	Generate random variates (samples)
`pdf(x)` / `pmf(k)`	Probability density / mass function
`cdf(x)`	Cumulative distribution function \(P(X \le x)\)
`sf(x)`	Survival function \(1 - \text{CDF}(x)\)
`ppf(q)`	Percent point function (inverse CDF)
`isf(q)`	Inverse survival function
`mean()`	Distribution mean
`var()`	Distribution variance
`std()`	Distribution standard deviation
`stats(moments)`	Central moments (mean, variance, skew, kurtosis)
`entropy()`	Differential entropy
`fit(data)`	Fit parameters to data (class method)
`interval(confidence)`	Confidence interval around the median

Continuous vs Discrete¶

Frozen distributions work identically for both continuous (rv_continuous) and discrete (rv_discrete) distributions. The only difference is that discrete distributions use .pmf() (probability mass function) instead of .pdf():

# Continuous: use pdf
normal = stats.norm(loc=0, scale=1)
normal.pdf(0)  # height of the density curve at x=0

# Discrete: use pmf
poisson = stats.poisson(mu=3.0)
poisson.pmf(3)  # exact probability P(X = 3)

Why Frozen Distributions Matter¶

Frozen distributions embody the object-oriented design of scipy.stats. Each distribution is an object that encapsulates its parameters and provides a unified API. This design pattern makes it easy to write generic code that works with any distribution, pass distributions as arguments to functions, and swap distributions in simulations without changing the calling code.

Summary¶

Frozen distributions are the standard way to work with scipy.stats. By fixing parameters at creation time, they provide a clean, consistent interface for sampling, density evaluation, probability computation, and statistical analysis.