Phenotypic resistance of indigenous goat breeds to infection with Haemonchus contortus in northwestern Nigeria

Computation of the Distribution of the Sum of Independent Negative Binomial Random Variables

“…The NB distribution modeled with the probability density function, 𝑁𝐵(𝜇, 𝜃), such that 𝜇 = 𝑚𝑒𝑎𝑛(𝑆 𝑛 ) and 𝜃 = 𝑚𝑒𝑎𝑛(𝑆 𝑛 ) 2 (𝑣𝑎𝑟(𝑆 𝑛 ) − 𝑚𝑒𝑎𝑛(𝑆 𝑛 )) ⁄ , better fits the exact distribution of the sum of NB variables, but still with a bias (Figure 1B). This confirms that the distribution of the sum of independent NB variables is not an NB, as wrongly stated in [6]. It is, rather, a mixture NB (see below) [9].…”

“…The sum of independent NB variables is of special interest in different contexts, such as the study of animal distribution [4,5], fecal egg counts in infected goats [6], the number of emergency medical calls [7], empirical distribution of the duration of wet periods in days [8] or insurance risk [9]. When the sum of several independent NB counts is available, determining the distribution of ∑ X i with X i ∼ NB(r i ; p i ) is a problem.…”

“…When working on the sum of variables, the first thought is to use the central limit theorem [13] that establishes that, in many situations, the distribution of the sum-independent random variables tends to go toward a normal distribution. An alternative is to model the distribution of the sum of NB variables, as an NB distribution is based on the observation that the distribution of the sum of NB variables is a mixture NB distribution [9], according to Theorem 2, proposed by Makun, Abdulganiyu, Shaibu, Otaru, Okubanjo, Kudi, and Notter [6].…”

“…. n, p j = j/10, and r j = j) as a function of the recursive iterations k from Equation (6). For the first eight iterations, the change in P(S 7 = 6) k is very small.…”