论文标题
指数随机变量，Hurwitz的Zeta功能和分区功能的最大最大变量
Maximum of Exponential Random Variables, Hurwitz's Zeta Function, and the Partition Function
论文作者
论文摘要
在优惠券收集器问题的背景下，自然问题是独立几何分布的随机变量（具有不同参数）的最大值的行为。 Brennan等人已经解决了这个问题。 （英国数学J.＆CS。8（2015），330-336）。在这里，我们为该最大值的矩以及具有相应参数的指数随机变量的最大值提供了显式渐近表达式。我们还处理每个变量是最大变量的概率。 计算导致表达式在某些特殊点上涉及Hurwitz的Zeta功能。我们在此处明确发现该点函数的值。同样，我们处理的最大值的分布函数与分区函数的生成函数密切相关。因此，我们的结果（和证明）依赖于与分区功能有关的经典结果。
A natural problem in the context of the coupon collector's problem is the behavior of the maximum of independent geometrically distributed random variables (with distinct parameters). This question has been addressed by Brennan et al. (British J. of Math. & CS. 8 (2015), 330-336). Here we provide explicit asymptotic expressions for the moments of that maximum, as well as of the maximum of exponential random variables with corresponding parameters. We also deal with the probability of each of the variables being the maximal one. The calculations lead to expressions involving Hurwitz's zeta function at certain special points. We find here explicitly the values of the function at these points. Also, the distribution function of the maximum we deal with is closely related to the generating function of the partition function. Thus, our results (and proofs) rely on classical results pertaining to the partition function.
