# Virtual University of Pakistan

## CS723 : Probability and Stochastic Processes

### Course Overview

#### Course Synopsis

This is a graduate level course. The course will start by presenting fundamental concepts of probability theory. It will then develop mathematically sound concepts of random variables and their processing through PDF and CDF.

#### Course Learning Outcomes

Upon successful completion of this course, students should be able to:

• Feel comfortable about concepts and terminology of probability theory and its domains of application.
• Apply set-theoretic probabilistic modeling of un-predictable phenomena and academic and real-life problems.
• Solve simple problems related to random variables, their distribution functions, expected values, moments, and their conditional expectations.
• Work with jointly distributed pairs of random variables using their joint and marginal densities.
• Understand how sequence of random variables behave and converge to predictable behaviour.

#### Course Calendar

 Topic Lecture Resource Page Introduction to Probability and Stochastic Processes 1 Papoulis (2002), Krishnan (2006) Set Theory 2 do. Relations, Functions, Probability 3 do. Probability Experiments: Repeated, Dependent, Cascaded 4 do. Joint Probability, Conditional Probability, Total Probability 5 do. Combinatronics, Birthday Problem, Bayes Rule 6 do. Bayes Rule Application, Partitioned Footpath, Partioned Floor, Trains on a Junction 7 do. Trains on a Junction, A local bus stop, Random line partition 8 do. Assignment No. 1 Equally likely outcomes, Random Variables, Chuk-a-luck, Binomial & Poisson Distributions 9 do. CDF of Poisson PMF, Joint Distribution Example, Marginal PMF 10 do. Conditional Distribution, Conditional PMF, Expected Value, Transformation of Random Variables 11 do. Transformation of RVs, Conditional Expectation, Co-variance and Correlation 12 do. Continuous Random Variable, Probability Density Function, PDF and CDF of a Continuous RV 13 do. CDF of a Continuous RV, CDF and PDF of Nozzle Height, Exponential RV 14 do. Failure of TV Set Example, Gaussian RV 15 do. Mean of Transformed RV, Height Distribution of Humans Example, Higher Moments 16 do. Assignment No. 2 Scaling of Gaussian RV, Standard Gaussian Events, Joint Distributions, Pair of RVs 17 do. Pair of RVs , Buffon's Needle, Exponential RV Pair, Pair of Erlang RVs 18 do. Exponential RV Pair, Pair of Dependent RVs, Correlation & Covariance 19 do. Correlation & Covariance, Correlated Gaussian RVs 20 do. Function of two RVs, Evaluation of Fz(Z), CDF & PDF of Z = X + Y 21 do. Function of two RVs, Evaluation of Fz(Z), CDF & PDF of Z = Y / X, CDF & PDF of Z = max (X, Y) 22 do. Mid-term Examination Multiple derived RVs, Probability Computation 23 do. Pair of derived RVs 24 do. Direct Computation of Derived CDF 25 do. Non-invertible Transformations 26 do. Moments of U and V from f(x, y) 27 do. Expectation of Transformed Random Variables, Covariance Matrix, Eigenvalues & Eigenvectors 28 do. Vector RVs, Transforming Vector RVs 29 do. PDF of Z = X+X ? 2X, Characteristic Functions 30 do. Convergence of Sequence, Convergence of RVs 31 do. Norm of Vectors and Function, Convergence of RVs, Inequalities 32 do. Basics of Markov chains, Examples of Markov chains 33 Lawler (2006) Transition probabilities, Stochastic matrix, Stationary probability distribution 34 do. Periodic and aperiodic Markov chains, Multivariate joint probability 35 do. Transient and recurrent Markov chains, Return times of recurrent states 36 do. Course Viva n-step stochastic matrix, n-step reachability 37 do. Diagonalization of stochastic matrix, Periodic and aperiodic recurrence 38 do. Academic Research Paper Eigen-analysis of stochastic matrix 39 do. Bipartite and tripartite graphs of Markov chains 40 do. Asymptotic behaviour of Markov chains 41 do. Partitioning of Markov chains 42 do. Presentations Markov chains with infinite state space 43 do. Random walk on real line, Random walk on 2-d plane, Random walk in higher dimensions, Continuous-time Markov chains 44 do. Continuous-time Markov chains, Poisson arrival and departure processes, Stationary probability distribution, Queuing theory 45 do. Final-term Examination