Exponential and Gamma¶

The exponential and gamma distributions are closely related continuous distributions that model waiting times and event durations. The exponential distribution models the time until a single event, while the gamma distribution generalizes this to the time until multiple events.

Exponential Distribution¶

The exponential distribution models the time between events in a Poisson process. Its PDF is:

where \(\lambda\) is the rate parameter (average number of events per unit time).

Parametrization in scipy.stats¶

An important detail: scipy.stats.expon uses the scale parameter, which is the reciprocal of the rate: \(\text{scale} = 1/\lambda\). When working with a rate \(\lambda\), you must pass scale=1/λ:

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

la = 3.0 # rate parameter λ a = stats.expon(scale=1/la) # scale = 1/λ

x = np.linspace(0, 3, 100) y_pdf = a.pdf(x) y_cdf = a.cdf(x)

plt.plot(x, y_pdf, label='PDF') plt.plot(x, y_cdf, label='CDF') plt.legend(loc='lower left') plt.title(f'Exponential Distribution (λ={la})') plt.xlabel('x') plt.show() ```

The PDF starts at its maximum value \(\lambda\) at \(x=0\) and decays exponentially. The CDF \(F(x) = 1 - e^{-\lambda x}\) rises from 0 toward 1.

Key Properties¶

The exponential distribution has mean \(E[X] = 1/\lambda\), variance \(\text{Var}(X) = 1/\lambda^2\), and the unique memoryless property: \(P(X > s + t \mid X > s) = P(X > t)\). This means that knowing the process has already lasted \(s\) time units provides no information about the remaining time.

Financial Applications¶

In finance, the exponential distribution models inter-arrival times of trades, time between defaults in credit risk, and duration analysis in survival models for corporate defaults.

Gamma Distribution¶

The gamma distribution is a generalization of the exponential. If \(X_1, \ldots, X_k\) are independent \(\text{Exp}(\lambda)\) random variables, then their sum follows a \(\text{Gamma}(k, 1/\lambda)\) distribution.

Parameters¶

```python import scipy.stats as stats

Gamma with shape=5, scale=2 (waiting for 5 events, mean inter-event time=2)¶

gamma_dist = stats.gamma(a=5, scale=2) print(f"Mean: {gamma_dist.mean():.2f}") # k * θ = 10 print(f"Variance: {gamma_dist.var():.2f}") # k * θ² = 20 ```

Special Cases¶

The gamma family includes several important special cases: \(\text{Gamma}(1, \theta) = \text{Exponential}(\text{scale}=\theta)\) and \(\text{Gamma}(k/2, 2) = \chi^2(k)\) (chi-square with \(k\) degrees of freedom).

Financial Applications¶

In finance, the gamma distribution is used in insurance claim modeling (aggregate loss distributions), Bayesian conjugate priors for Poisson rate estimation, and stochastic volatility models where variance follows a gamma process.

Summary¶

The exponential and gamma distributions form a natural family for modeling durations and waiting times. The key practical point when using scipy.stats is the scale parametrization: always pass scale=1/λ when working with rate parameters.

Exercises¶

Exercise 1. Verify the memoryless property of the exponential distribution numerically. For \(X \sim \text{Exp}(\lambda=1)\), compute \(P(X > 3 \mid X > 1)\) using the survival function and compare it to \(P(X > 2)\).

from scipy import stats

rv = stats.expon(scale=1)
p_conditional = rv.sf(3) / rv.sf(1)
p_direct = rv.sf(2)
print(f"P(X>3 | X>1) = {p_conditional:.6f}")
print(f"P(X>2)       = {p_direct:.6f}")
print(f"Equal: {abs(p_conditional - p_direct) < 1e-10}")

Exercise 2. Show that the sum of \(n\) independent \(\text{Exp}(\lambda)\) random variables follows a \(\text{Gamma}(n, 1/\lambda)\) distribution. Generate 10,000 sums of 5 exponential samples with \(\lambda = 2\) and compare the sample mean/variance with \(\text{Gamma}(5, 0.5)\) theoretical values.

import numpy as np
from scipy import stats

np.random.seed(42)
exp_samples = stats.expon.rvs(scale=0.5, size=(10000, 5))
sums = exp_samples.sum(axis=1)

gamma_rv = stats.gamma(a=5, scale=0.5)
print(f"Sample mean: {np.mean(sums):.4f}, Theoretical: {gamma_rv.mean():.4f}")
print(f"Sample var:  {np.var(sums, ddof=1):.4f}, Theoretical: {gamma_rv.var():.4f}")

Exercise 3. Fit a gamma distribution to 500 samples generated from \(\text{Gamma}(a=4, \text{scale}=3)\). Use stats.gamma.fit() with floc=0 and compare the estimated parameters with the true values.

import numpy as np
from scipy import stats

np.random.seed(42)
data = stats.gamma.rvs(a=4, scale=3, size=500)
a_hat, loc_hat, scale_hat = stats.gamma.fit(data, floc=0)
print(f"Estimated a={a_hat:.4f} (true 4), scale={scale_hat:.4f} (true 3)")