Bounds for the difference between median and mean of beta and negative binomial distributions (Q917141)

This paper uses the technique developed by \textit{J. Chen} and \textit{H. Rubin} [see Stat. Probab. Lett. 4, 281-283 (1986; Zbl 0596.62015)] to obtain bounds on the difference between the mean and median of the beta distribution and then to solve the same problem for the negative binomial distribution. Theorem 1. Let X have Beta\((p,q)\) distribution. Then \(0<E X-\)median\((X)<A(p,q)=(q-p)/(p+q-2)(p+q)\), \(1<p<q\), \(0<E X-\)median\((X)<A(q,p)\), \(1<q<p,\) and \(E X=\) median\((X)\), \(p=q.\) Theorem 2. Let X have negative bionomial \((r,p)\) distribution. Then \(1<E X-\)median\((X)<(1-p)/p\), \(p<1/2,\) \((1-p)/p<E X-\)median\((X)<1\), \(p>1/2,\) and \(E X=\)median\((X)\), \(p=1/2.\)