| Abstract: | Consider a sequence of independent and identically distributed random variables with an absolutely continuous distribution function. The probability functions of the numbers Kn,r and Nn,r of r-records up to time n of the first and second type, respectively, are obtained in terms of the non central and central signless Stirling numbers of the first kind. Also, the binomial moments of Kn,r and Nn,r are expressed in terms of the non central signless Stirling numbers of the first kind. The probability functions of the times Lk,r and Tk,r of the kth r-record of the first and second type, respectively, are deduced from those of Kn,r and Nn,r. A simple expression for the binomial moments of Tk,r is derived. Finally, the probability functions and binomial moments of the kth inter-r-record times Uk,r = Lk,r ? Lk?1,r and Wk,r = Tk,r ? Tk?1,r are obtained as sums of finite number of terms. |
