Log Pdf: Nadar
First, compute the normalizer: ( -\ln(1-0.8) = -\ln(0.2) = 1.60944 )
This write-up explores the mathematical foundation, key properties, applications, and generation of the Probability Density Function (PDF) for the Nadar Log distribution. The Nadar Log distribution is a discrete distribution (support ( k = 1, 2, 3, \dots )) whose probability mass function is proportional to a logarithmic series. The standard form of its PDF (or more accurately, its Probability Mass Function, since it's discrete) is given by:
import numpy as np import matplotlib.pyplot as plt def nadar_log_pmf(k, theta): """Compute PMF for Nadar Log distribution.""" norm = -np.log(1 - theta) return (theta**k) / (k * norm) nadar log pdf
theta = 0.7 k_values = np.arange(1, 21) pmf_values = nadar_log_pmf(k_values, theta)
[ P(X = k) = \frac\theta^k-k \ln(1-\theta), \quad k = 1, 2, 3, \dots ] First, compute the normalizer: ( -\ln(1-0
Understanding this distribution equips data scientists and statisticians with another lens through which to view and model real-world count data.
plt.stem(k_values, pmf_values) plt.title(f'Nadar Log PDF (θ = theta)') plt.xlabel('k') plt.ylabel('P(X=k)') plt.grid(alpha=0.3) plt.show() The Nadar Log PDF (Logarithmic distribution) is a discrete, heavy-tailed probability model derived directly from the logarithmic series. Its key characteristics—mode at 1, overdispersion, and polynomial tail decay—make it a powerful tool for modeling rare event counts in ecology, linguistics, and beyond. While less common than the normal or Poisson distributions, it occupies a critical niche for data where small values dominate but large values occur more frequently than exponential models would predict. [ -\ln(1-\theta) = \theta + \frac\theta^22 + \frac\theta^33
[ -\ln(1-\theta) = \theta + \frac\theta^22 + \frac\theta^33 + \dots = \sum_k=1^\infty \frac\theta^kk ]
In the vast landscape of probability distributions, some are celebrated for modeling natural phenomena (like the Normal distribution), while others serve highly specialized niches. The Nadar Log PDF (often referred to in literature as the Log-Nadarajah distribution or simply the Logarithmic distribution) falls into the latter category. It is a compelling example of a discrete probability distribution derived from a logarithmic series, with unique properties that make it invaluable in specific fields like ecology, linguistics, and information theory.
