A useful lemma and its application to negative binomial distribution. (English) Zbl 0792.62011
The paper considers certain relationships between the parameters and median of a negative binomial distribution. The nice result of this paper is as follows:
In a negative binomial distribution with parameters \(K\) and \(\theta\) the non-negative integer \(K\) satisfies \[ \sum_{z=0}^ K {{z+k-1} \choose {k-1}} \theta^ k (1-\theta)^ z= {1/2} \quad \text{if and only if} \quad K=k-1 \quad \text{and} \quad \theta=1/2. \] {}.
In a negative binomial distribution with parameters \(K\) and \(\theta\) the non-negative integer \(K\) satisfies \[ \sum_{z=0}^ K {{z+k-1} \choose {k-1}} \theta^ k (1-\theta)^ z= {1/2} \quad \text{if and only if} \quad K=k-1 \quad \text{and} \quad \theta=1/2. \] {}.
Reviewer: N.G.Gamkrelidze (Moskva)
MSC:
| 62E10 | Characterization and structure theory of statistical distributions |
| 60E05 | Probability distributions: general theory |
