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Course Information
Title
Theory of Probability and Mathematical Expectations
Theory Credits
3 (3 hrs/week)
Practical Credits
1 (2 hrs/week)
Course Outcomes
After successful completion of the course, students will be able to:
Acquaint with the role of statistics in dealing with the univariate random variables.
Learn the extension of the univariate data to bivariate data.
Learn the measure of randomness mathematically by using expectations.
Get familiarity about generating functions, law of large numbers and the central limit theorem,
further to apply in research and allied fields.
Theory — Five Units
Unit 1: Elementary Probability
Basic Concepts of Probability, random experiments, trial, outcome, sample space, event, mutually
exclusive and exhaustive events, equally likely and favourable outcomes. Mathematical, Statistical,
axiomatic definitions of probability. Conditional Probability and independence of events, Addition
and multiplication theorems of probability for 2 and for n events and simple problems. Boole's
inequality, Bayes' theorem and its applications in real life problems.
Definition of random variable (r.v.), discrete and continuous random variables, functions of
random variable. Probability mass function, Probability density function, Distribution function and
its properties. Calculation of moments, coefficient of skewness and kurtosis for a given pmf and pdf.
Mathematical expectation of function of a random variable. Moments and covariance using
mathematical expectation with examples. Addition and Multiplication theorems on expectation.
Properties of expectations, variance, covariance. Chebyshev and Cauchy-Schwartz inequalities and
their applications.
Definitions of Moment Generating Function, Cumulant Generating Function, Characteristic Function
and Probability Generating Function and their properties. Weak Law of Large Numbers (WLLN), Strong
Law of Large Numbers (SLLN). Convergence in probability and convergence in distribution, concept of
Central limit theorem.