Course Overview
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Course Synopsis
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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.
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Course Learning Outcomes
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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.
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Course Calendar
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1
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Introduction to Probability and Stochastic Processes
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3
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Relations, Functions, Probability
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4
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Probability; Experiments: Repeated, Dependent, Cascaded
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5
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Joint Probability, Conditional Probability, Interesting Events, Total Probability
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Academic Research Paper
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6
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Combinatronics, Birthday Problem, Bayes Rule
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7
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Bayes Rule: Application, Partitioned Footpath, Partioned Floor, Trains on a Junction, . .
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8
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Trains on a Junction, A local bus stop, Random line partition
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9
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Equally likely outcomes, Random variables, Chuk-a-luck, Binomial & Poisson Distributions,
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10
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CDF of Poisson PMF, Joint Distribution: Example, Marginal PMF
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Assignment No. 1
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11
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Conditional Distribution, Conditional PMF, Expected Value, Transformation of RV’s
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12
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Transformation of RV’s, Conditional Expectation, Co-variance / Correlation
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13
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Continuous Random Variable, Probability Density Function, PDF CDFof a Continuous RV, . . .
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14
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CDF of a Continuous RV, CDF PDF of Nozzle Height, Exponential RV
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15
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Failure of TV Set, Gaussian RV
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16
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Mean of Transformed RV, Height Distribution of Humans, Higher Moments, . . .
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17
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Scaling of Gaussian RV, Standard Gaussian Events, Joint Distributions, Pair of RVs, . . .
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18
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Pair of RVs, Buffon's Needle, Exponential RV Pair, Pair of Erlang RVs
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Mid-term Examination
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19
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Exponential RV Pair, Pair of Dependent RVs, Correlation & Covariance, . . .
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20
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Correlation & Covariance, Correlated Gaussian RVs
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Assignment No. 2
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21
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Function of two RVs, Evaluation of Fz(Z), CDF / PDF of Z = X + Y, . . .
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22
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Function of two RVs, Evaluation of Fz(Z), CDF / PDF of: Z = Y / X, Z = max (X, Y), . . .
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23
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Multiple derived RVs, Probability Computation, . . .
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25
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Direct computation of derived CDF
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26
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Non-invertible Transformations
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27
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Moments of U and V from f(x, y)
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28
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Expectation of transformed random variables, Covariance matrix, Eigenvalues & eigenvectors
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29
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Vector RVs, Transforming Vector RVs
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30
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PDF of Z = X+X ? 2X, Characteristic Functions, . . .
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Course Viva
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31
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Convergence of Sequence, Convergence of RVs
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32
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Norm of Vectors and Function, Convergence of RV's, Inequalities
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34
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Markov Chains, Total Probability
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Academic Term Paper Presentation
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36
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Markov Chains, A few Definitions
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41
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Markov Chains (Cont.).
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42
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Markov Chains (Cont.)..
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43
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Markov Chains (Cont.)...
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44
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Markov Chains (Cont.)....
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45
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Markov Chains (Cont.).....
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Final-term Examination
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