In this article, new properties of the Poisson distribution of order k with parameter λ are found. Based on them, the modes of the Poisson distributions of order k=3 and 4 are derived for λ in (0,1). They are 0, 3, 5, and 0, 4, 7, 8, respectively, for λ in specified subintervals of (0, 1). In addition, using Mathematica, computational results for the modes of the Poisson distributions of order k=2,3, and 4 are presented for λ in specified subintervals of (0,2).

The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind 2F1(z) are included in all connection coefficients for a specific z. Several new connection formulae between some famous polynomials, such as Fibonacci, Lucas, Pell, Fermat, Pell–Lucas, and Fermat–Lucas polynomials, are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed.

One of the most widely known and important applications of probability and statistics is scientific polling to forecast election results. In 1936, Gallup predicted correctly the victory of Roosevelt over Landon in the US presidential election, using scientific sampling of a few thousand persons, whereas the Literary Digest failed using 2.4 million answers to 10 million mailed questionnaires to automobile and telephone owners. Since then, polls have grown to be a multibillion flourishing and very influential and important industry, spreading around the world. Polls have mostly been accurate in the US presidential elections, with a few exceptions. Their two most notable failures were their wrong predictions of the US 1948 and 2016 presidential elections. Most polls failed too in the 2016 UK Referendum, in the 2014 and 2019 India Lok Sabha elections, and in the US 2020 presidential election, even though in the latter three they did predict the winner. We discuss these polls in the present paper. The failure in 1948 was due to non-random sampling. In 2016 and 2020 it was mainly due to the problem of non-response and possible biases of the pollsters. In 2014 and 2019 it was due to non-response and political biases of the polling agencies and news outlets that produced the polls.

Upper and lower bounds are derived for the mode(s) of the negative binomial distribution of order k, type I, with parameters r and p, which are employed to establish an explicit formula for the mode(s) in terms of r and k when p equals 0.5. It is also shown as a direct consequence of the upper bound alone that the mode is k when r equals 1. The derivation of the bounds is based on a known recurrence relation satisfied by the probability mass function of the distribution.

During the last decade, many researchers have focused on proving identities that reveal the relation between Fibonacci and Lucas numbers. Very recently, one of these identities has been generalized to the case of Fibonacci and Lucas numbers of order k. In the present work, we state and prove a new identity regarding an alternating sum of Fibonacci and Lucas numbers of order k. Our result generalizes recent works in this direction.

In the present paper we introduce a modified consecutive system, which generalizes consecutive systems extensively studied in the literature, and we obtain its reliability when its components are functioning independently with probabilities not necessarily equal. The results are illustrated by numerical examples using MATLAB and potential applications are discussed.

Let F-n((k)) = 0 for + -k +1 <= n <= 0, F-1((k)) = 1, and F-n((k)) = Sigma(k)(j=1) F-n-j((k)) for n >= 2. Also let L-0((k)) = k, L-1((k)) = 1, L-n((k)) = n + Sigma(n-1)(j=1) L-n-j((k)) n for 2 <= n <= k, and L-n((k)) = Sigma(k)(j=1) L-n-j((k)) for n >= k+1. The identity Sigma(n)(i=0) m(i) ((L-i((k)) + (m - 2)F-i+1((k)) - Sigma(k)(j=3) (j - 2) F-i-j+1((k)))) = m(n+1) F-n+1((k)) + k - 2 (m >= 2; k >= 2); derived recently by means of colored tiling [4], is presently proved using only the definitions of F-n((k)) and L-n((k)), and the identity L-n((k)) = Sigma(k)(j=1) j F-n-j+1((k)) (n >= 1).

The following relation between Fibonacci and Lucas numbers of order k,∑i=0nmi[li(k)+(m−2)Fi+1(k)−∑j=3k(j−2)Fi−j+1(k)]=mn+1Fn+1(k)+k−2, is derived by means of colored tiling. This relation generalizes the well-known Fibonacci-Lucas identities, ∑i=0n2iLi=2n+1Fn+1,∑i=0n3i(Li+Fi+1)=3n+1Fn+1 and ∑i=0nmi(Li+(m−2)Fi+1)=mn+1Fn+1 of A.T. Benjamin and J.J. Quinn, D. Marques, and T. Edgar, respectively.

This article is an attempt to generalize some of the recent papers on randomized response techniques by using the negative binomial distribution of order k to randomize the responses in the randomization design where respondents can report outcome of one of two binary devices depending upon their actual status. The relative efficiency results are observed to be better than those of many recent and relevant randomized response techniques. The results are also better than those of the base line model used in this study, providing the sensitive attribute is rare. An extra advantage of the proposed technique is that it does not require any additional sampling and administrative cost.

For integers m >= 0 and k >= 2, set alpha(m,k) := Sigma(infinity)(n-1) n(m) F-n((k))/2(n+1-1) where F-n((k)) is the Fibonacci sequence of order k or k -generalized Fibonacci sequence. It is shown that alpha(0,k) = 1, alpha(1,k) = 2(k+1) - k - 1, alpha(2),(k) = 2(k+1) (2(k+2) - 4k - 3) + k(2) + 2k -1, and alpha(m,k) = 1 + Sigma(m-1)(r=0) ((m)(r)) Sigma(k)(i=1) 2(k-i) i(m-r) alpha(r,k) which generalize recent results on weighted Fibonacci sums by Benjamin, Neer, Otero, and Sellers.

Sharp upper and lower bounds are established for the modes of the Poisson distribution of order k. The lower bound established in this paper is better than the previously established lower bound. In addition, for k = 2, 3, 4, 5, a recent conjecture is presently proved solving partially an open problem since 1983.

In the present paper we deal with two difierent problems concerning engineering, the mov- ing window detection problem and weaning from me- chanical ventilation. We review the research on these two areas and contribute to it by studying waiting time distributions of patterns of two successes sepa- rated by at most or exactly k i 2 failures (k ‚ 3) in the case of flrst-order dependent trials.

Let Z 1, Z 2, . . . be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Z t  = 1) and q = Pr(Z t  = 0) = 1 − p, respectively, t = 1, 2, . . . . For any given integer k ≥ 2 we consider the patterns \({\mathcal{E}_{1}}\): two successes are separated by at most k−2 failures, \({\mathcal{E}_{2}}\): two successes are separated by exactly k −2 failures, and \({\mathcal{E}_{3}}\) : two successes are separated by at least k − 2 failures. Denote by \({ N_{n,k}^{(i)}}\) (respectively \({M_{n,k}^{(i)}}\)) the number of occurrences of the pattern \({\mathcal{E}_{i}}\) , i = 1, 2, 3, in Z 1, Z 2, . . . , Z n when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let \({T_{r,k}^{(i)}}\) (resp. \({W_{r,k}^{(i)})}\) be the waiting time for the r − th occurrence of the pattern \({\mathcal{E}_{i}}\), i = 1, 2, 3, in Z 1, Z 2, . . . according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study of \({N_{n,k}^{(i)}}\), \({M_{n,k}^{(i)}}\), \({T_{r,k}^{(i)}}\) and \({W_{r,k}^{(i)}}\) (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.

In the present paper we study the waiting time distributions of patterns of two successes separated by at most or exactly k-2 failures (k >= 3) in the case of first-order dependent trials. Employing both non-overlapping and overlapping counting schemes we obtain closed formulas for the probability generating functions and effective recursive schemes for the evaluation of the probability mass functions of the waiting time random variables. Finally, we give applications of our results to the moving window detection problem and a biomedical engineering one.

Let Z1,Z2,… be a sequence of Bernoulli trials with success probability p=Pr(Zt=1) and failure probability q=Pr(Zt=0)=1−p, t⩾1. For positive integers k1 and k2 we consider the events E1: at least k1 consecutive 0's are followed by at least k2 consecutive 1's, E2: exactly k1 consecutive 0's are followed by exactly k2 consecutive 1's and E3: at most k1 consecutive 0's are followed by at most k2 consecutive 1's. Denote by Xn(i) the number of occurrences of the event Ei(i=1,2,3) in Z1,Z2,…,Zn(n⩾1), and let Tr(i) be the waiting time for the r-th occurrence of the event Ei(i=1,2,3) in Z1,Z2,…. In the present paper we employ the Markov chain embedding technique to derive exact formulas for the probability generating functions, the probability mass functions and the m-th moments (m⩾1) of Xn(i) and Tr(i)(i=1,2,3). An application is also given.

There are several well-known formulas counting the number of distinct allocations of n indistinguishable objects into m distinguishable cells, each of which has capacity k - 1. In the present paper we generalize four of them by relaxing the assumption that each of the m cells has capacity k - 1 and assuming instead that there are s kinds of cells and each cell of kind i has capacity k(i) - 1 (i = 1, ..., s). A generalization of the Pascal triangles of order k is also discussed.

Statistics denoting the numbers of success runs of length exactly equal and at least equal to a fixed length, as well as the sum of the lengths of success runs of length greater than or equal to a specific length, are considered. They are defined on both linearly and circularly ordered binary sequences, derived according to the Pólya-Eggenberger urn model. A waiting time associated with the sum of lengths statistic in linear sequences is also examined. Exact marginal and joint probability distribution functions are obtained in terms of binomial coefficients by a simple unified combinatorial approach. Mean values are also derived in closed form. Computationally tractable formulae for conditional distributions, given the number of successes in the sequence, useful in nonparametric tests of randomness, are provided. The distribution of the length of the longest success run and the reliability of certain consecutive systems are deduced using specific probabilities of the studied statistics. Numerical examples are given to illustrate the theoretical results.

The Polya-Eggenberger sampling scheme is considered, i.e., a ball is drawn at random from an urn containing w white (success) balls and b black (failure) balls, its color is observed, and then it is returned to the urn along with s additional balls of the same color of the ball drawn. This scheme is repeated n times, and N-n,N-k,N-l,N-s denotes the number of l-overlapping success runs of length k. If the balls drawn are arranged on a circle, the number of I-overlapping success runs of length k is denoted by N-n,k,l,s(c).. Finally, W-r,W-k,W-l,W-s denotes the number of drawings according to the Polya-Eggenberger sampling scheme until the rth occurrence of the l-overlapping success run of length k. Polya, inverse Polya, and circular Polya distributions of order k for l-overlapping success runs of length k are introduced as the distributions, respectively, of N-n,N-k,N-l,N-s, W-r,W-k,W-l,W-s, and N-n,k,l,s(c). These distributions include as special cases known and new distributions of order k. Exact formulae are derived for their probability distribution functions and means, which generalize several results on well-known distributions of order k. Asymptotic results of the new distributions are also given, relating them, respectively, to the binomial, negative binomial, and circular binomial distributions of order k for l-overlapping success runs of length k.