Exact Distributions of the Number of r-Records and the r-Record and Inter-r-Record Times

摘要

Consider a sequence of independent and identically distributed random variables with an absolutely continuous distribution function. The probability functions of the numbers K_(n,r) and N_(n,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 K_(n,r) and N_(n,r) are expressed in terms of the non central signless Stirling numbers of the first kind. The probability functions of the times L_(k,r) and T_(k,r) of the kth r-record of the first and second type, respectively, are deduced from those of K_(n,r) and N_(n,r). A simple expression for the binomial moments of T_(k,r) is derived. Finally, the probability functions and binomial moments of the kth inter-r-record times U_(k,r) = L_(k,r) - L_(k-1,r) and W_(k,r) = T_(k,r) - T_(k-1,r) are obtained as sums of finite number of terms.