# A note on the modes of the negative binomial distribution of order k, type I

## TL;DR

This paper derives bounds and explicit formulas for the modes of the negative binomial distribution of order k, type I, providing insights into its behavior for specific parameters, especially when p equals 0.5.

## Contribution

It introduces new bounds and an explicit mode formula for the negative binomial distribution of order k, type I, based on recurrence relations.

## Key findings

- Bounds for the modes are established using recurrence relations.
- Explicit mode formula derived for p=0.5.
- Mode equals k when r=1 based on bounds.

## Abstract

Upper and lower bounds are derived for the mode(s) of the negative binomial distribution of order k, type I, with parameters r and p, which are employed to establish an explicit formula for the mode(s) in terms of r and k when p equals 0.5. It is also shown as a direct consequence of the upper bound alone that the mode is k when r equals 1. The derivation of the bounds is based on a known recurrence relation satisfied by the probability mass function of the distribution.

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Source: https://tomesphere.com/paper/1702.02183