[ P(X = k) = \frac\theta^k-k \ln(1-\theta), \quad k = 1, 2, 3, \dots ]

[ -\ln(1-\theta) = \theta + \frac\theta^22 + \frac\theta^33 + \dots = \sum_k=1^\infty \frac\theta^kk ]

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)

First, compute the normalizer: ( -\ln(1-0.8) = -\ln(0.2) = 1.60944 )

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.

theta = 0.7 k_values = np.arange(1, 21) pmf_values = nadar_log_pmf(k_values, theta)