Syllabus (COMPUTER APPLICATION)

Course Type: ME-4

Semester: 4

Course Code: BBCAMEA24T

Course Title: Probability and Statistics

(L-P-Tu): 4-0-0

Credit: 4

Practical/Theory: Theory

Course Objective: Course Objectives: 1. Students will use appropriate statistical terms to describe data. 2. Student will use appropriate statistical methods to collect, organize, display, and analyze relevant data. 3. Students will compute fluently and make reasonable estimations. 4. Students will apply basic concepts of probability. 5. Students will apply concepts of various probability distributions to find probabilities.

Learning Outcome: Course Outcomes: Upon successful completion of this course, students will be able to- 1. Calculate the expectation and moments of one and two dimensional random variables. 2. Use of some important one dimensional discrete and continuous distributions and their basic properties. 3. Learn the concept correlation and regression. 4. Explain the concept of convergence and check for the convergence of a given sequences of random variables. 5. Find the expressions for the characteristic function of a random variable and verify its properties. 6. Apply the various laws of large numbers to sequences of random variables. 7. Understand the basic components of sampling and have the knowledge on exact sampling distributions which are essential for estimating and testing hypothetical statements. 8. Find a best estimator with reference the different criteria in case of real-life applications. Understand critically the problems that are faced in testing of a hypothesis. 9. Apply the different testing tools like t-test, chi-square test etc. to analyze the real-life problems.

Syllabus:

Unit -1: Probability [Credit: 2, 30L]

Sample space and Axiomatic definition of probability, Compound experiment.

One and two dimensional random variables (Discrete and Continuous): Distribution function and its basic properties, Probability density function, Marginal distribution and density function, Conditional density function.

Transformation of one and two dimensional random variables.

Mathematical expectation, Median, Mode, Moments, Variance.

Expectation for two dimensional case, Moments, Covariance, Correlation coefficient and its properties, Addition and multiplication rule for expectation and variance, Independent random variables. Moment generating function, Characteristic function.

Conditional expectation and regression – Least square regression lines and basic properties.

Some important distributions: Binomial, Poisson, Uniform, exponential, Cauchy, Normal, Gamma, Beta and their basic property.

Tchebycheff’s inequality, Convergence in probability, Bernoulli’s limit theorem, Central limit theorem.

Approximations – Binomial to Poisson, Central limit theorem (statement only). Binomial to Normal (De Moivre-Laplace limit theorem).

Unit -2: Statistics [Credit: 2, 30L]

Concept of statistics, Sampling distribution of sample mean for finite population with examples.

Sampling distribution for infinite population, Exact sampling distribution-for mean of normal population, Chi-square and t-distribution.

Point Estimation – consistent, unbiased, MVUE. Method of maximum likelihood and application to different population.

Interval estimation- Method of finding confidence interval- Application to normal population.

Testing of hypothesis: Critical region, Type-I and Type-II error, Power of test, Large sample test related to Binomial proportion, Chi-square test on multinomial distribution, Exact tests for mean and variance of univariate Normal distribution.

Reading References:

1. A.P. Baisnab and M. Jas; Elements of Probability and Statistics ; Tata McGraw Hill Co. Ltd
2. Amritava Gupta; Groundwork of Mathematical Probability and Statistics; Academic
3. Publishers.
4. Gun, Gupta, Dasgupta; Fundamentals of Statistics; World Press.
5. S.C. Gupta, V .K. Kapoor; Fundamentals of Mathematical Statistics; S Chand & Sons.
6. N.G. Das; Statistical Methods; M Das & Co.