Sampling (rvs)

The .rvs() method generates random variates (random samples) from a probability distribution. It is the primary tool for Monte Carlo simulation, bootstrapping, and any workflow that requires synthetic data from a known distribution.

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Basic Usage

Every frozen distribution in scipy.stats provides the .rvs() method:

import scipy.stats as stats

a = stats.norm(loc=3.0)  # frozen normal distribution, mean=3, std=1
samples = a.rvs(size=(2, 3), random_state=1)
print(samples)
# [[4.62434536 2.38824359 2.47182825]
#  [1.92703138 3.86540763 0.6984613 ]]
print(type(samples))   # <class 'numpy.ndarray'>
print(samples.shape)   # (2, 3)
print(samples.dtype)   # float64

The output is always a NumPy array whose shape is determined by the size parameter. With size=(2, 3), you get a 2x3 matrix of independent samples.

Parameters

Parameter	Type	Description
size	int or tuple	Shape of the output array. size=1000 gives a 1D array; size=(100, 5) gives a 2D array.
random_state	int, Generator, or RandomState	Seed for reproducibility. Pass an integer for deterministic results.

Reproducibility

The random_state parameter ensures that the same sequence of random numbers is generated each time:

# These two calls produce identical samples
s1 = stats.norm(0, 1).rvs(size=5, random_state=42)
s2 = stats.norm(0, 1).rvs(size=5, random_state=42)
assert (s1 == s2).all()

For more control, pass a numpy.random.Generator object:

import numpy as np
rng = np.random.default_rng(seed=42)
samples = stats.norm(0, 1).rvs(size=1000, random_state=rng)

Verifying Samples Against Theory

A standard validation technique is to overlay a histogram of samples with the theoretical PDF (continuous) or PMF (discrete):

import scipy.stats as stats
import numpy as np
import matplotlib.pyplot as plt

# Generate samples and overlay with theoretical PDF
a = stats.norm(loc=0, scale=1)
samples = a.rvs(size=10000, random_state=337)
x = np.linspace(-4, 4, 100)

plt.hist(samples, density=True, bins=50, alpha=0.7, label='Histogram')
plt.plot(x, a.pdf(x), 'r-', linewidth=2, label='Theoretical PDF')
plt.legend()
plt.title('Sampling Verification: Histogram vs PDF')
plt.show()

As the sample size increases, the histogram converges to the theoretical distribution — a visual demonstration of the law of large numbers.

Sampling from Different Distribution Types

The .rvs() method works identically for continuous and discrete distributions:

# Continuous distributions
stats.norm(0, 1).rvs(size=5)           # normal
stats.expon(scale=1/3).rvs(size=5)     # exponential with rate λ=3

# Discrete distributions
stats.poisson(mu=3.0).rvs(size=5)      # Poisson
stats.binom(n=100, p=0.6).rvs(size=5)  # binomial

For discrete distributions, .rvs() returns integer-valued arrays.

Financial Applications

Random sampling is fundamental to Monte Carlo methods in finance: simulating asset price paths under geometric Brownian motion, generating scenarios for Value at Risk (VaR) estimation, pricing path-dependent options, and stress testing portfolio performance under various distributional assumptions.

Summary

The .rvs() method is the gateway to simulation-based analysis. Combined with the random_state parameter for reproducibility and NumPy's array infrastructure, it provides an efficient and consistent interface for generating random samples from any scipy.stats distribution.