introduction to Probability, Statistics and Random Processes

Description

To expose the students to the modern theory
of probability, concept of random variables
and their expectations.
To introduce various discrete and continuous
distributions and concept of estimation
theory, confidence interval.
To illustrate the concept of hypothesis
testing, tests for means and variances,
Goodness of fit tests.
To introduce the concept of random
processes, Markov chains, Brownian
Motion.
Course Outcomes
Define and apply the concepts of probability
and conditional probability.
Define and illustrate discrete and continuous
random variables, their probability mass
functions and probability density functions.
Understand the concept and need of
hypothesis testing.
Perform the tests for means and variances
and Goodness of fit test.
Understand the concept of random
processes, Markov chains, Brownian
motions.
Syllabus
Axiomatic construction of the theory of
probability, independence, conditional
probability, and basic formulae.
Random variables and distributions: Univariate,
Bivariate and multivariate random variables,
Cumulative and marginal distribution function,
Conditional and multivariate distributions,
Functions of random variables: Sum, product,
ratio, change of variables.
Mathematical expectations, moments, moment
generating function, characteristic functions;
Discrete/continuous distributions and limit
theorems: Binomial distribution, Geometric
distribution, Poisson distribution, Normal
distribution, Exponential distribution, Gamma
distribution, Beta distribution, Central limit
theorem, Chebyshev’s inequality, Law of large
numbers.
Estimation Theory: Bias of estimates,
Confidence intervals, Minimum variance
unbiased estimation, Bayes’ estimators, Moment
estimators, Maximum likelihood estimators,
Chi-square distribution, Confidence intervals for
parameters of normal distribution.
Hypothesis testing: Tests for means and
variances, hypothesis testing and confidence
intervals, Bayes’ decision rules, Power of tests,
Goodness-of-fit tests, Kolmogorov-Smirnov
Goodness-of-fit test.
Definition and classification of random
processes, discrete-time Markov chains, Poisson
process, continuous-time Markov chains,
stationary processes, Gaussian process,
Brownian motion.
Textbooks/ References
1. S. Ross, Introduction to Probability and
Statistics for and Engineers and Scientists,
Third Edition, Elsevier, 2004.
2. P. G. Hoel, S. C. Port and C. J. Stone,
Introduction to Probability Theory, Universal
Book Stall, 2000.
3. S. M. Ross, Introductory Statistics, Second
Edition, Academic Press, 2009.
4. J. Medhi, Stochastic Processes, Third Edition,
New Age International, 2009.
5. V.K.Rohati and A.K. Saleh, An introduction
to Probability and Statistics, Third Edition.
Wiley Student Edition, 2006.
6. G. R. Grimmett and D. R. Stirzaker,
Probability and Random Processes, Oxford
University Press, 2001.
7. W. Feller, An Introduction to Probability
Theory and its Applications, Vol. 1, Third
Edition.,Wiley, 1968.
8. S.M. Ross, Stochastic Processes, Second
Edition. Wiley, 1996.
9. C. M. Grinstead and J. L. Snell, Introduction
to Probability, Second Edition, Universities
Press India, 2009