Pakistan experiences the southwest monsoon normally from July to September every year. In this paper, we analyse the effect of monthly rainfall for the months of July, August and September on overall seasonal monsoon for the period of 1960–2006 across 37 different meteorological stations of Pakistan. The statistical results are based on the notion of conditional probability. We find conditional probabilities of seasonal monsoon to be in excess, deficient and normal when the monthly rainfall of July, August and September is considered to be in excess and deficient. When rainfall in the months of July and August is in excess the seasonal monsoon behaviour is also likely to be in excess and probabilities of seasonal monsoon to be deficient are zero for most of the meteorological stations. Similarly, when rainfall in the months of July and August is considered to be deficient the seasonal monsoon behaviour is also likely to be deficient and probabilities of seasonal monsoon to be in excess are zero for most of the meteorological stations. © 2013 Royal Meteorological Society

In this paper, a comprehensive empirical study is conducted to evaluate the performance of a new real-coded crossover operator called Fisk crossover (FX) operator. The basic aim of the proposed study is to preserve population diversity as well as to avoid local optima. In this context, a new crossover operator is designed and developed which is linked with Log-logistic probability distribution. For its global performance, a realistic comparison is made between FX versus double Pareto crossover (DPX), Laplace crossover (LX), and simulated binary crossover (SBX) operators. Moreover, these crossover operators are also used in conjunction with three mutation operators called power mutation (PM), Makinen, Periaux, and Toivanen mutation (MPTM), and nonuniform mutation (NUM) for inclusive evaluation. The performance of probabilistic-based algorithms is tested on a set of twenty-one well-known nonlinear optimization benchmark functions with diverse features. The empirical results show a substantial dominance of FX over other crossover operators with authentication of performance index (PI). Moreover, we also examined the significance of the proposed crossover scheme by administrating ANOVA and Gabriel pairwise multiple comparison test. Finally, the statistically significant results of the proposed crossover scheme have a definite edge over the other schemes, and it is also expected that FX has a great potential to solve complex optimization problems.

In the present simulation-based study, a novel parent-centric real-coded crossover operator is introduced with a unique probabilistic aspect of the mixture distribution. Moreover, the mixture distribution is a co-integration of double Pareto and Laplace probability distributions with various parameters. The key objective of the newly proposed methodology is to obtained optimal solutions for complex multimodal optimization problems. Hence, for its global comparison, the newly proposed mixture distribution crossover operator (MDX) is compared with double Pareto (DPX), Laplace (LX), and simulated binary (SBX) crossover operators within the conjunction of three mutation operators (MTPM, PM, and NUM). After a descriptive comparison, a Quade multiple comparison test is also administered to examine its statistical significance. Furthermore, the performance of the genetic algorithm (GA) is also examined on a set of twenty-one unconstraint benchmark functions with diverse features. The empirical results of the simulation-based study reveal that the mixture-based crossover operator obtained a substantial dominance over all considered crossover operators in terms of computational complexity, robustness, scalability, and capability of exploration and exploitation. Moreover, the Quade multiple comparison test also showed a significant superiority with graphical authentication of the performance index (PI).

In the context of the classical two-player gambler's ruin problem, the winning probabilities and initial stakes are pre-decided. If a player (who is in financial crisis) starts with less amount than his/her opponent in the symmetric game, has more chances to be ruined. Besides, a player (based on previous record data) with more winning probability than his/her competitor, has fewer chances to be ruined. We observe that most of the time, usually a weaker player is not fully willing to make a contest with a strong player. To give a fair chance to fight back for a weaker player and to develop the audience's interest, equity-based modeling is required. In this research, we propose some new equity-based models for the game of two players. In this way, we advocate the weaker player (with less winning probability or less amount to start the game) is motivated to participate in the contest because of a fair chance to make a comeback. The working methodology of newly proposed schemes is executed by deriving general expressions of the ruin probabilities for mathematical evaluation along with observing the ruin times, and then findings are compared with the results of a classic two-player game. Hence, the prime objectives related to the study are achieved by taking diverse parametric settings in the favor of equity-based modeling.