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2 Probability and Mathematical Statistics

2 2023

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New easy to compute formulas for the moments of random variables appearing in the coupon collectorproblem

León-García Amaryani, Pérez Aroldo, Bolívar-Cimé Addy

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 2

155--164

Assuming that there are N types of coupons, where the prob- ability that the ith coupon appears is pi ≥ 0 for i = 1, . . . , N , with [formula], we consider the variable Tk which represents the number of acquisitions needed to obtain k ≤ N different coupons, and the variable Yn which represents the number of different coupons obtained in n acquisitions. In the coupon collector problem it is of interest to obtain the expected value of these random variables, as well as their rth moments. We provide new expressions for the rth moments of Tkand Yn, and we give expressions for their moment generating functions. Unlike known formulas, our formula for the rth moment of Tk is given in terms of recursive expressions and that of Yn is given in terms of finite sums, so that they can be easily implemented computationally. Furthermore, our formulas allow obtaining simplified expressions of the first few moments of the variables.

Moment inequalities for nonnegative random variables

Castañeda-Leyva Netzahualcóyotl, Rodríguez-Narciso Silvia, Hernández-Quintero Angélika, Pérez Aroldo

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 1

41--61

We give reciprocal versions of the Sclove et al. and Feller inequalities for moments of nonnegative random variables. Our results apply to any nonnegative random variable. The strongest assumption is that the moments involved must be finite. Thus, the results obtained also hold for any empirical distribution with nonnegative data. These facts allow potential applications in numerical analysis, probability, and statistical inference, among other disciplines. Moreover, the proposed methodology offers an alternative approach to obtain new inequalities and even to improve some known inequalities. For instance, we give new inequalities for the ratio of gamma functions. In this context, we also improve an inequality by Bustoz and Ismail and some cases of inequalities due to Gurland and Dragomir et al. Additionally, we present a new inequality for finite sums of nonnegative or nonpositive numbers. For some cases, this relation improves even the Cauchy-Bunyakovsky-Schwarz inequality.