# Negative Binomial Distribution Essay

BY

OWUSU BRIGHT.

INTRODUCTION

In centuries, human activities where outcomes are described either as success or failure, has been a concern in the history of probability and statistics. In the act of tossing a coin, throwing a die to obtain a 6, either a machine work or fails, a student passing an exam or not and so on are events whose end results are either a yes or no, good or bad, present or absent, success or failure, as well as a win or loss. Generally, the history of probability and statistics, describe these activities to follow a Bernoulli distribution of the discrete probability distributions where an experiment has only two possible outcomes with a

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Linden & Samu. Mantyniemi, 1415). However, the process of parameterization of the negative binomial distribution employed by the author resulted in the distribution having a number of failures (X) before the r^th Success with independent trials where a probability of success ‘p’ was defined for each trial, i.e. (0≤p≤1) and the probability mass function associated with the distribution was defined

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Consequently, the process of parameterization was repeated several times to obtain different parameters of ‘r’ and ‘p’. “These newly generated parameters of the negative binomial distribution were then substituted into Eqn. 1 to obtain maximum likelihood estimations where curves were then fitted to the migration data of the Woodlark birds”. (A. Linden & Samu. Mantyniemi, 1415). A gradual change in migration over the season was modeled to estimate the expected number of migrants.

The author then provided the plot below to describe the negative binomial and overdispersion for the expectation of the migration curve. The plot of the graph A, described the Poisson modeling process for counts of the birds which shows many birds that were excluded from the count due to the variation in the migration of the birds. Nevertheless, Plot B, C, and D represents the negative binomial distribution model with linear and quadratic mean – variance relationship where less or no bird was left uncounted in the migration