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In this paper, there is the attempt of examining the long run steady state market shares of a variety of soap in Sambalpur, Odisha by applying the Markovian theory. The matrix consists of possibilities of transition of consumers who is switching different brand of bathing soap. The Markovian approach gives an insight of the reason of customer switching and try to identify the potential completion of a brand in the market and predicts the future condition of gaining and losing the customers of a particular brand. All the calculations done by primary data and using this data probability matrix has been designed.

Keywords: - Brand, Switching, Bath soap, share, Markov chain.

Fertility and population estimation techniques are examined in this study. Numerous countries undergo significant demographic changes that could significantly impact future fertility and demographic growth rates. the expanding population and fertility rates of a specific nation are the main topic of this article’s analysis of some demographic trends. In the future, statistical models must account for measurement error and uncertainty when making predictions to achieve the highest degree of accuracy for demographic research. Fertility, which we assess as "Fertility Measures," is one of the key elements influencing a place's natural population expansion. The world's level of fertility varies greatly depending on the country, culture, socioeconomic status, and age of the individual. Calculations of population predictions depict the population's future growth under specific assumptions about how it will change in the future, typically to fertility, migration, and mortality. For this analysis, Denmark, a northern European nation, is considered. By the recent dataset of Denmark’s population of the 4th quarter by age and live births (on the first day of the quarter), we also estimate the population projections by using the year 2011 as a base year to forecast the population growth of 2025. We concluded that total fertility had increased in 2021, and we predict the population forecast in 2025.

Keywords: Fertility Measures, Population Projection, Growth Models, Demographic Trends, Estimation.

The international demand for banana is an opportunity for India to increase its export. The data on export quantity and value of banana in India from the period of 1990-91 to 2017-18 were considered with the aim to carry out time series modeling for Indian banana export and value and to forecast the Indian banana export and its value for next five years. The Indian banana with respect of growth in quantity exhibited as positive growth rate of 22.14 per cent annum. The quantum of banana exported exhibited less variability with coefficient of variation at 121.32 per cent while it was maximum in the banana value during study period with coefficient of variation at 146.14 per cent. The instability index of banana export and value was 38.85 and 48.53 per cent respectively. The forecasted banana export and their value revealed rising trend. It was predicted that banana export are estimated to increase from 1.01 lakh tons in 2017 to 1.32 lakh tons in 2023. Similarly export value from banana will raise from Rs. 34,877 lakh in 2017 to Rs. 57,815 lakh. Hence, it is suggested that there is a need to give more concentration on export. ARIMA (3,1,6) and Brown’s exponential smoothing model was found best fit for banana export and its total value.

Key words: Banana, Export, Price, Instability, Forecasting

The sequence of all positive integers arranged in increasing order is called as the sequence of Natural Numbers. In this paper, we model the probability distribution for the second diagonal elements of Pascal’s triangle obtained from the sequence of natural numbers as the first diagonal elements. Explicit functional relation for the p.m.f. of the said distribution has been formulated. Functional forms of different statistical measures have been derived from the developed p.m.f. of the mentioned distribution. This study also derived the other properties like mean, variance, Pearson’s coefficient, moment generating functions, characteristic functions etc. In order to understand the model behaviour, numerical illustrations have been applied, from which the statistical measures are obtained from the derived mathematical formulae. The study further expanded by providing the complete statistical analysis with the suitable data visualization tools. This study has good scope and applicability in different areas such as architects, graphic designers, banking, etc.

Keywords: Natural Number’s Sequence, Pascal’s Triangle, Combinatorial Probability Distribution, Characteristic Function, Pearson’s Coefficients.

A Square is a quadrilateral with four sides and equal angles. The diagonals of a square are equal and bisect each other. The step-formed square is also a square with different unit lengths. In this paper, we model the probability distribution of a sequence of overlapping squares within a step-formed square in the case of odd-numbered courts on the baseline. We explored the direct functional relationships for the formulated distribution's probability mass function (PMF). We have verified the regulations of the probability mass function (PMF). This study has focused on deriving different mathematical relations of the statistical measures for location, scaling, shaping, peakedness etc. Further, we discussed the inter-probability distribution properties, generating functions, characteristic functions, etc. Sensitivity analysis uses suitable numerical illustrations to understand the model behaviour. This study has numerous real-time applications in the context of Combinatorics.

Keywords: Discrete probability distribution, squares within a step formed Squares, Pearson’s coefficients, sensitivity analysis.