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Annexure ‘CD – 01’ Course Title: Statistical Methods and Probability Distribution Course Code: MSA 102 Credit Units: 03 # 1 2 3 L T P/S SW/FW 03 - - - Course Title Course Objectives: To familiarize students with the basic procedures for summarization, analysis and presentation of statistical data and also introduce probability concepts, characteristics of the basic univariate distribution, joint and conditional distribution, correlation, regression and statistical software SPSS. Pre-requisites: Student should have basic knowledge of probability, statistics and calculus. Student Learning Outcomes: On completion the student will be able to: (1) Summarize the main features of a data set. (2) Explain and apply concepts of probability, conditional probability and independence of events. (3) Specify the meaning and application of a probability distribution and the distribution function of a random variable, the expected value of a random variable and the expectation of functions of a random variable for both a discrete and continuous random variable. (4) Recognize and evaluate features of some basic standard discrete and continuous distribution and their features. (5) Generate values of a random variable from a specified distribution using simulation methods. (6) Write and make use of joint distribution and marginal and conditional distribution, distribution of one or more linear combinations of several random variables. TOTAL CREDIT UNITS 03 Comments (if any) (7) Describe and explain the meaning of co-variation and evaluate strength of correlation between variables, Use of least square regression models, evaluate coefficient of determination, Describe/explain multiple linear regression, multiple and partial correlation. Course Contents/Syllabus: Module I: Data Collection, Presentation and Summarization Data type Primary vs. secondary Qualitative (attributive) vs. quantitative; nominal, ordinal, interval and ratio measurement scales Presentation of Data : Tabulation-Frequency Distribution, Frequency Tables Cumulative Frequency Distribution Graphical Methods and : Charts Histogram and cumulative frequency curves Diagrammatic presentation (ogives) Stem-leaf and Box-plot, Graphical presentation of time series data Measures of Location : Mean, median and mode, quartiles Measures of Spread : Range, Semi-inter quartile range, mean deviation, standard deviation, coefficient of variation Measures of Skewness : Symmetry and skewness, kurtosis; measures based on moments and other measures. Module II: Probability Concepts Basic elements of probability: sample space, simple events, mutually exclusive events, addition rule of probability, conditional probability definition, independence of events, multiplication rule of probability, law of total probability, Bayes’ theorem. Module III: Random Variables, Expectation & Generating Functions Random variables Distributions of random variables and their probability functions. Probability mass function (pmf) and cumulative distribution function (cdf) for a discrete random variable. Probability density function (pdf) and cumulative distribution function (cdf) for a continuous random variable; Expected value of a random variable. Comments (if any) 5% Weightage 10% Weightage 20% Weightage Expected value of a function of a random variable; Properties of an expected value, variance and moments of the distribution of a random variable; factorial moment of discrete random variables. Generating functions: Probability generating function for a discrete random variable, factorial – moment generating function, moment generating function and cumulant generating function for a random variable. Markov inequality and Chebychev Inequality Module IV: Univariate Distributions Discrete distributions: Uniform, Bernoulli, Binomial, Geometric, Negative binomial, Hyper geometric, Poisson (with Poisson process postulates); other discrete distributions (beta-binomial and logarithmic); truncated and censored random variables. Continuous distributions: Uniform (Rectangular), Normal, Exponential, Gamma, Beta; Lognormal, Pareto, Weibull, Burr, Laplace (double- exponential); truncated distributions Module V: Joint Distributions & Conditional Distributions Jointly distributed discrete and continuous random variables, joint probability (mass and density) functions, Joint cumulative distribution function (bivariate case) marginal distributions, Marginal density function, marginal pmf, Marginal pdf, marginal cdf, Multinomial distribution, bivariate normal distribution, Conditional distributions, conditional pmf, conditional pdf, conditional cdf. Stochastic independence of random variable-discrete and continuous case; random variables uncorrelated .Expectation of uncorrelated functions of jointly distributed random variables, covariance and correlation coefficient. Conditional expectation and conditional variance. Expectation of product of functions of independent random variables, Joint moment generating functions and moments; Properties for expectation, moments and mgf. Module VI: Correlation and Regression Correlation: Simple, partial and multiple correlation measures. Linear regression models, Response Variable and Covariates, Linear prediction of response variable, least squares fitting method, residuals; Partition of variation of response variable into “explained” and “unexplained” parts, coefficient of determination; multiple linear regression model and its fitting; examination of residuals and model diagnostics; step-up regression method. Module VII: Simulation 20% Weightage 20% Weightage 10% Weightage 5% Weightage General techniques for simulating continuous random variables The Inverse transformation Method The Rejection Method The Hazard Rate Method Special techniques for simulating continuous random variables. 1. Normal distribution, Gamma distribution, Chi –Square distribution, a. Variance Reduction techniques 1. Use of Antithetic Variables 2. Variance of Reduction by conditional Monte Carlo 3. Control Variates 4. Importance sampling Module VIII: Computer application of probability distribution using SPSS Managing Data: Listing cases, replacing missing values, computing new variables, recording variables, exploring data, selecting cases, sorting cases, merging files, Listing data Graphs: Creating and editing graphs and charts Frequencies: Frequencies, bar charts, histograms, percentiles Descriptive Statistics: measures of central tendency, variability, deviation from normality, size and stability. Cross Tabulation and chi-square analyses The means Procedure Bivariate Correlation: Bivariate Correlation, Partial Correlations and the correlation matrix Regression: Simple Linear Regression, Multiple regression analysis 10% Weightage Pedagogy for Course Delivery: Lectures Assignments Practicals Assessment/ Examination Scheme: Theory L/T (%) 100% Lab/Practical/Studio (%) End Term Examination 100% Theory Assessment (L&T): Continuous Assessment/Internal Assessment Components (Drop down) End Term Examination Class Test Assignment Practical Attendance 10% 5% 10% 5% Weightage (%) 30% Text & References: John E Freund’s Mathematical Statistics with Applications: Irwin Miller and Marylees Miller, Pearson Education Modern Mathematical statistics: E.J. Dudewicz and S.N. Mishra,Wiley Business Statistics:J.K.Sharma, Pearson Education Remarks and Suggestions: __________________________________ Date: Name, Designation, Organisation