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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.

On the longest runs in Markov chains

Liu Z., Yang X.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 2

407--428

In the first n steps of a two-state (success and failure) Markov chain, the longest success run L(n) has been attracting considerable attention due to its various applications. In this paper, we study L(n) in terms of its two closely connected properties: moment generating function and large deviations. This study generalizes several existing results in the literature, and also finds an application in statistical inference. Our metod on the moment generating function is based on a global estimate of the cumulative distribution function of L(n) proposed in this paper, and the proofs of the large deviations include the Gärtner-Ellis theorem and the moment generating function.