Comment on Aljohani et al. The Uniform Poisson–Ailamujia Distribution: Actuarial Measures and Applications in Biological Science. Symmetry 2021, 13, 1258

In this note, we would like to point out that the uniform Poisson–Ailamujia introduced by Aljohani et al. is a reparametrized geometric distribution.

Characterization of the Geometric Distribution Via Linear Combinations of Observations and of Records

In a sequence of independent identically distributed geometric random variables, the sum of the first two record values is distributed as a simple linear combination of geometric variables. It is verified that this distributional property characterizes the geometric distribution. A related characterization conjecture is also discussed. Related discussion in the context of weak records is also provided.

Kinetic characteristics of transcriptional bursting in a complex gene model with cyclic promoter structure

While transcription occurs often in a bursty manner, various possible regulations can lead to complex promoter patterns such as promoter cycles, giving rise to an important issue: How do promoter kinetics shape transcriptional bursting kinetics? Here we introduce and analyze a general model of the promoter cycle consisting of multi-OFF states and multi-ON states, focusing on the effects of multi-ON mechanisms on transcriptional bursting kinetics. The derived analytical results indicate that bust size follows a mixed geometric distribution rather than a single geometric distribution assumed in previous studies, and ON and OFF times obey their own mixed exponential distributions. In addition, we find that the multi-ON mechanism can lead to bimodal burst-size distribution, antagonistic timing of ON and OFF, and diverse burst frequencies, each further contributing to cell-to-cell variability in the mRNA expression level. These results not only reveal essential features of transcriptional bursting kinetics patterns shaped by multi-state mechanisms but also can be used to the inferences of transcriptional bursting kinetics and promoter structure based on experimental data.

Characterizations of Twenty (2020-2021) Proposed Discrete Distributions

In this paper, certain characterizations of twenty newly proposed discrete distributions: the discrete gen- eralized Lindley distribution of El-Morshedy et al.(2021), the discrete Gumbel distribution of Chakraborty et al.(2020), the skewed geometric distribution of Ong et al.(2020), the discrete Poisson X gamma distri- bution of Para et al.(2020), the discrete Cos-Poisson distribution of Bakouch et al.(2021), the size biased Poisson Ailamujia distribution of Dar and Para(2021), the generalized Hermite-Genocchi distribution of El-Desouky et al.(2021), the Poisson quasi-xgamma distribution of Altun et al.(2021a), the exponentiated discrete inverse Rayleigh distribution of Mashhadzadeh and MirMostafaee(2020), the Mlynar distribution of Fr¨uhwirth et al.(2021), the flexible one-parameter discrete distribution of Eliwa and El-Morshedy(2021), the two-parameter discrete Perks distribution of Tyagi et al.(2020), the discrete Weibull G family distribution of Ibrahim et al.(2021), the discrete Marshall–Olkin Lomax distribution of Ibrahim and Almetwally(2021), the two-parameter exponentiated discrete Lindley distribution of El-Morshedy et al.(2019), the natural discrete one-parameter polynomial exponential distribution of Mukherjee et al.(2020), the zero-truncated discrete Akash distribution of Sium and Shanker(2020), the two-parameter quasi Poisson-Aradhana distribution of Shanker and Shukla(2020), the zero-truncated Poisson-Ishita distribution of Shukla et al.(2020) and the Poisson-Shukla distribution of Shukla and Shanker(2020) are presented to complete, in some way, the au- thors’ works.

Fitting of a Markov chain model for daily rainfall data at Calcutta

A Markov chain probability model has been fitted to the daily rainfall data recorded at Calcutta. The 'wet spell' and 'weather cycles' are found to obey geometric distribution, The distribution of the number of rainy days per week has been calculated and compared with the actual data.

Does a Starting Positive End-Expiratory Pressure of 8 cmH2O Decrease the Probability of a Ventilator-Associated Event?

Introduction: Ventilator-associated events (VAEs) are objective measures as defined by the Centers for Disease Control and Prevention (CDC). To reduce VAEs, some hospitals have started patients on higher baseline positive end-expiratory pressure (PEEP) to avoid triggering VAE criteria due to respiratory fluctuations.Methods: At our institution, VAEs were gathered from January 2014 through December 2019. Using the CDC-defined classifications, VAEs were split into two groups to separate patients with hypoxemia only (VAC) and those with hypoxemia and evidence of inflammation or infection (IVAC-plus). We used the geometric distribution to calculate the daily event probability before and after the protocol implementation. A probability threshold was used to determine if the days between events was exceeded during the post-protocol period.Results: A total of 306 VAEs were collected over the study period. Of those, 155 were VACs and 107 were IVAC-plus events during the pre-protocol period. After implementing the protocol, 24 VACs and 20 IVAC-plus events were reported. There was a non-significant decrease in daily event probabilities in both the VAC and IVAC-plus groups (0.083 vs. 0.068 and 0.057 vs. 0.039, respectively).Conclusion: We concluded a starting PEEP of 8 cmH2O is unlikely to be an effective intervention at reducing the probability of a VAE. Until specific guidelines by the CDC are established, hospitals should consider alternative methods to reduce VAEs.

Zip Trees

We introduce the zip tree , 1 a form of randomized binary search tree that integrates previous ideas into one practical, performant, and pleasant-to-implement package. A zip tree is a binary search tree in which each node has a numeric rank and the tree is (max)-heap-ordered with respect to ranks, with rank ties broken in favor of smaller keys. Zip trees are essentially treaps [8], except that ranks are drawn from a geometric distribution instead of a uniform distribution, and we allow rank ties. These changes enable us to use fewer random bits per node. We perform insertions and deletions by unmerging and merging paths ( unzipping and zipping ) rather than by doing rotations, which avoids some pointer changes and improves efficiency. The methods of zipping and unzipping take inspiration from previous top-down approaches to insertion and deletion by Stephenson [10], Martínez and Roura [5], and Sprugnoli [9]. From a theoretical standpoint, this work provides two main results. First, zip trees require only O (log log n ) bits (with high probability) to represent the largest rank in an n -node binary search tree; previous data structures require O (log n ) bits for the largest rank. Second, zip trees are naturally isomorphic to skip lists [7], and simplify Dean and Jones’ mapping between skip lists

On the Discrete Weibull Marshall–Olkin Family of Distributions: Properties, Characterizations, and Applications

In this article, we introduce a new flexible discrete family of distributions, which accommodates wide collection of monotone failure rates. A sub-model of geometric distribution or a discrete generalization of the exponential model is proposed as a special case of the derived family. Besides, we point out a comprehensive record of some of its mathematical properties. Two distinct estimation methods for parameters estimation and two different methods for constructing confidence intervals are explored for the proposed distribution. In addition, three extensive Monte Carlo simulations studies are conducted to assess the advantages between estimation methods. Finally, the utility of the new model is embellished by dint of two real datasets.