Lognormal and Weibull¶

The lognormal and Weibull distributions are both continuous distributions on \((0, \infty)\) commonly used for modeling positive, right-skewed data. The lognormal distribution arises from multiplicative processes and is central to financial asset price modeling. The Weibull distribution is the standard model for reliability analysis and survival times, with a shape parameter that controls whether the failure rate increases, decreases, or remains constant over time.

Lognormal Distribution¶

A random variable \(Y\) follows a lognormal distribution if \(\ln Y\) is normally distributed. Formally, if \(X \sim N(\mu, \sigma^2)\), then \(Y = e^X \sim \text{Lognormal}(\mu, \sigma^2)\).

PDF¶

The probability density function is:

where \(\mu\) and \(\sigma\) are the mean and standard deviation of the underlying normal distribution \(\ln Y\), not of \(Y\) itself.

Parametrization in scipy.stats¶

An important detail: scipy.stats.lognorm uses the parameter s for \(\sigma\) and scale for \(e^\mu\). To create a lognormal distribution with parameters \(\mu\) and \(\sigma\), pass s=σ and scale=exp(μ):

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

mu, sigma = 0.5, 0.8 a = stats.lognorm(s=sigma, scale=np.exp(mu))

print(f"Mean: {a.mean():.4f}") # exp(μ + σ²/2) print(f"Variance: {a.var():.4f}")

x = np.linspace(0, 10, 200) 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() plt.title(f'Lognormal Distribution (μ={mu}, σ={sigma})') plt.xlabel('y') plt.ylabel('Density / Probability') plt.show() ```

The PDF is right-skewed with a peak below the mean, reflecting the asymmetry inherent in the exponential transformation.

Key Properties¶

• Mean: \(E[Y] = e^{\mu + \sigma^2/2}\)
• Variance: \(\text{Var}(Y) = (e^{\sigma^2} - 1)\,e^{2\mu + \sigma^2}\)
• Median: \(e^\mu\) (the median equals the scale parameter in scipy)
• Mode: \(e^{\mu - \sigma^2}\)

Financial Applications¶

The lognormal distribution is central to quantitative finance. The Black-Scholes model assumes that stock prices follow a geometric Brownian motion, which implies that prices at any future time are lognormally distributed. It is also used for modeling income distributions, insurance claim sizes, and asset values in credit risk models.

Weibull Distribution¶

The Weibull distribution is a flexible model for lifetime data and reliability analysis. Its PDF is:

where \(k > 0\) is the shape parameter and \(\lambda > 0\) is the scale parameter.

Parametrization in scipy.stats¶

The scipy.stats.weibull_min function uses c for the shape parameter \(k\) and scale for \(\lambda\):

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

k, lam = 2.0, 5.0 a = stats.weibull_min(c=k, scale=lam)

print(f"Mean: {a.mean():.4f}") print(f"Variance: {a.var():.4f}")

x = np.linspace(0, 15, 200) 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() plt.title(f'Weibull Distribution (k={k}, λ={lam})') plt.xlabel('x') plt.ylabel('Density / Probability') plt.show() ```

Effect of the Shape Parameter¶

The shape parameter \(k\) controls the failure rate behavior, making the Weibull distribution versatile for reliability modeling:

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

x = np.linspace(0, 3, 200) for k in [0.5, 1.0, 1.5, 2.0, 3.0]: plt.plot(x, stats.weibull_min(c=k, scale=1).pdf(x), label=f'k={k}')

plt.legend() plt.title('Weibull PDFs for Various Shape Parameters (λ=1)') plt.xlabel('x') plt.ylabel('f(x)') plt.show() ```

Key Properties¶

• Mean: \(E[X] = \lambda\,\Gamma(1 + 1/k)\)
• Variance: \(\text{Var}(X) = \lambda^2\left[\Gamma(1 + 2/k) - \Gamma(1 + 1/k)^2\right]\)

Special Cases and Failure Rate¶

• \(k = 1\): reduces to the exponential distribution with rate \(1/\lambda\) (constant failure rate)
• \(k < 1\): decreasing failure rate (infant mortality or burn-in phase)
• \(k > 1\): increasing failure rate (aging or wear-out phase)
• \(k = 2\): the Rayleigh distribution (after appropriate scaling)

Parameters in scipy.stats¶

Financial Applications¶

In finance, the Weibull distribution models time-to-default for bonds, operational risk event durations, and equipment failure in physical asset valuation. It is also used in insurance for modeling claim development periods and policy lapse rates.

Summary¶

The lognormal and Weibull distributions are both flexible models for positive, right-skewed data. The key practical point for scipy.stats is the parametrization: use stats.lognorm(s=σ, scale=exp(μ)) for lognormal and stats.weibull_min(c=k, scale=λ) for Weibull.

Exercises¶

Exercise 1. Generate 5000 samples from a lognormal distribution with \(\mu = 0\) and \(\sigma = 0.5\) (in log-space). Compute the sample mean and compare with the theoretical mean \(e^{\mu + \sigma^2/2}\).

import numpy as np
from scipy import stats

rv = stats.lognorm(s=0.5, scale=np.exp(0))
samples = rv.rvs(size=5000, random_state=42)
theo_mean = np.exp(0 + 0.5**2 / 2)
print(f"Sample mean: {np.mean(samples):.4f}")
print(f"Theoretical mean: {theo_mean:.4f}")

Exercise 2. Plot the Weibull PDF for shape parameters \(c = 0.5, 1.0, 2.0, 5.0\) on the same axes. Identify which shapes correspond to decreasing, constant, and increasing hazard rates.

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

x = np.linspace(0.01, 3, 200)
for c in [0.5, 1.0, 2.0, 5.0]:
    plt.plot(x, stats.weibull_min.pdf(x, c), label=f'c={c}')
plt.legend()
plt.title('Weibull PDF for Various Shape Parameters')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.ylim(0, 3)
plt.show()
# c < 1: decreasing hazard, c = 1: constant, c > 1: increasing

Exercise 3. Fit a lognormal distribution to the following data: take 1000 samples from \(\text{Lognormal}(\mu=2, \sigma=0.8)\), use stats.lognorm.fit(), and recover the underlying \(\mu\) and \(\sigma\) parameters. Recall that scipy parametrizes lognormal with shape \(s = \sigma\), scale \(= e^\mu\).

import numpy as np
from scipy import stats

np.random.seed(42)
data = stats.lognorm.rvs(s=0.8, scale=np.exp(2), size=1000)
s_hat, loc_hat, scale_hat = stats.lognorm.fit(data, floc=0)
mu_hat = np.log(scale_hat)
print(f"Estimated sigma (s): {s_hat:.4f} (true: 0.8)")
print(f"Estimated mu: {mu_hat:.4f} (true: 2.0)")