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We introduce the new continuous distribution, the so-called beta-Lindley distribution that extends the Lindley distribution. We provide a comprehensive mathematical treatment of this distribution. We derive the moment generating function and therth moment thus, generalizing some results in the literature. Expressions for the density, moment generating function, andrth moment of the order statistics also are obtained. Further, we also discuss estimation of the unknown model parameters in both classical and Bayesian setup. The usefulness of the new model is illustrated by means of two real data sets. We hope that the new distribution proposed here will serve as an alternative model to other models available in the literature for modelling positive real data in many areas.

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In this article, we analyze the growth pattern of Covid-19 pandemic in India from March 4th to July 11th using regression analysis (exponential and polynomial), auto-regressive integrated moving averages (ARIMA) model as well as exponential smoothing and Holt-Winters smoothing models. We found that the growth of Covid-19 cases follows a power regime of (t^2,t,..) after the exponential growth. We found the optimal change points from where the Covid-19 cases shift their course of growth from exponential to quadratic and then from quadratic to linear. After that, we saw a sudden spike in the course of the spread of Covid-19 and the growth moved from linear to quadratic and then to quartic, which is alarming. We have also found the best fitted regression models using the various criteria such as significant p-values, coefficients of determination and ANOVA etc. Further, we search the best fitting ARIMA model for the data using the AIC (Akaike Information Criterion) and provide the forecast of Covid-19 cases for future days. It was noticed that the ARIMA model fits better the Covid-19 cases for small regions. ARIMA (5, 2, 5) and ARIMA (5, 2, 3) are the best possible models for modeling Covid-19 cases for March 4th to July 10th and June 1th to July 10th, respectively.

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Nicotine, primary component of tobaco produces craving and withdrawal effect both in humans and animals. Nicotine shows a close resemblance to other addictive drugs in molecular, neuroanatomical and pharmacological, particularly the drugs which enhances the cognitive functions. Nicotine mainly shows its action through specific nicotinic acetylcholine receptors located in brain. It stimulates presynaptic acetylcholine receptors thereby enhancing Ach release and metabolism. Dopaminergic system is also stimulated by it, thus increasing the concentration of dopamine in nuclear accumbens. This property of nicotine according to various researchers is responsible for reinforcing behavioral change and dependence of nicotine. Various researchers have also depicted that some non dopaminergic systems are also involved for rewarding effect of nicotinic withdrawal. Neurological systems such as GABAergic, serotonergic, noradrenergic, and brain stem cholinergic may also be involved to mediate the actions of nicotine. Further, the neurobiological pathway to nicotine dependence might perhaps be appropriate to the attachment of nicotine to nicotinic acetylcholine receptors, peruse by stimulation of dopaminergic system and activation of general pharmacological changes that might be responsible for nicotine addiction. It is also suggested that MAO A and B both are restrained by nicotine. This enzyme helps in degradation dopamine, which is mainly responsible for nicotinic actions and dependence. Various questions remain uninsurable to nicotine mechanism and require more research. Also, various genetic methods united with modern instrumental analysis might result for more authentic information for nicotine addiction.

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We have developed the Bayesian estimation procedure for flexible Weibull distribution under Type-II censoring scheme assuming Jeffrey's scale invariant (noninformative) and Gamma (informative) priors for the model parameters. The interval estimation for the model parameters has been performed through normal approximation, bootstrap, and highest posterior density (HPD) procedures. Further, we have also derived the predictive posteriors and the corresponding predictive survival functions for the future observations based on Type-II censored data from the flexible Weibull distribution. Since the predictive posteriors are not in the closed form, we proposed to use the Monte Carlo Markov chain (MCMC) methods to approximate the posteriors of interest. The performance of the Bayes estimators has also been compared with the classical estimators of the model parameters through the Monte Carlo simulation study. A real data set representing the time between failures of secondary reactor pumps has been analysed for illustration purpose.