Course11110211192 lectures
Introduction to Probability Theory and Stochastic Processes
Prof. S Dharmaraja · IIT Delhi
About
This course explanations and expositions of probability and stochastic processes concepts which they need for their experiments and research. It also covers theoretical concepts of probability and stochastic processes pertaining to handling various stochastic modeling. This course provides random variable, distributions, moments, modes of convergences, classification and properties of stochastic processes, stationary processes, discrete and continuous time Markov chains and simple Markovian queueing models.
INTENDED AUDIENCE : Under-graduate students of electrical engineering, computer engineering, mechanical engineering, civil engineering and mathematics and computing
PREREQUISITES : A basic course on Calculus and Linear Algebra
INDUSTRY SUPPORT : Fractal Analytics, Genpact, Goldman Sachs, FinMechanics, Deutsche Bank and other finance companies.
Lectures
Week 1
- Random experiment, sample space, axioms of probability, probability space
- Random experiment, sample space, axioms of probability, probability space. (Cont...)
- Random experiment, sample space, axioms of probability, probability space. (Contd...)
- Conditional probability, independence of events.
- Multiplication rule, total probability rule, Bayes's theorem.
Week 2
- Definition of Random Variable, Cumulative Distribution Function
- Definition of Random Variable, Cumulative Distribution Function (Continued 1)
- Definition of Random Variable, Cumulative Distribution Function (Continued 2)
- Type of Random Variables, Probability Mass Function, Probability Density Function
- Type of Random Variables, Probability Mass Function, Probability Density Function (continued 1)
- Distribution of Function of Random Variables
Week 3
- Mean and Variance
- Mean and Variance Continued
- Higher Order Moments and Moments Inequalities
- Higher Order Moments and Moments Inequalities Continued
- Generating Functions
- Generating Functions Continued
Week 4
- Common Discrete Distributions
- Common Discrete Distributions Continued
- Common Continuous Distributions
- Common Continuous Distributions Continued
- Applications of Random Variable
- Applications of Random Variable Continued
Week 5
- Random vector and joint distribution
- Joint probability mass function
- Joint probability density function
- Independent random variables
- Independent random variables Continued
Week 6
- Functions of several random variables
- Functions of several random variables continued
- Some important results
- Order statistics
- Conditional distributions
- Random sum
Week 7
- Moments and Covariance
- Variance Covariance matrix
- Multivariate Normal distribution
- Probability generating function and Moment generating function
- Correlation coefficient
- Conditional Expectation
- Conditional Expectation continued
Week 8
- Modes of Convergence
- Mode of Convergence (Continued)
- Law of Large Numbers
- Central Limit Theorem
- Central Limit Theorem (Continued)
Week 9
- Motivation for Stochastic Processes
- Definition of a Stochastic Process
- Classification of Stochastic Processes
- Examples of Stochastic Process
- Examples Of Stochastic Process (Continued)
- Bernoulli Process
- Poisson Process
- Poisson Process (Continued)
- Simple Random Walk
- Time Series and Related Definitions
- Strict Sense Stationary Process
- Wide Sense Stationary Process and Examples
- Examples of Stationary Processes Continued
Week 10
- Discrete Time Markov Chain (DTMC)
- DTMC continued
- Examples of DTMC
- Examples of DTMC continued
- Chapman-Kolmogorov equations and N-step transition matrix
- Examples based on N-step transition matrix
- Examples continued
- Classification of states
- Classification of states continued
- Calculation of N-Step-9
- Calculation of N-Step-10
- Limiting and Stationary distributions
- Limiting and Stationary distributions continued
Week 11
- Continuous time Markov chain (CTMC)
- CTMC continued
- State transition diagram and Chapman-Kolmogorov equation
- Infinitesimal generator and Kolmogorov differential equations
- Limiting distribution
- Limiting and Stationary distributions -1
- Birth death process
- Birth death process continued
- Poisson process -1
- Poisson process continued
- Poisson process (cont.)
- Non-homogeneous and compound Poisson process
Week 12
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IIT Delhi
- Department
- Mathematics
- Teacher
- Prof. S Dharmaraja
- Level
- Undergraduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Teens profiles and older, judged from what the source says about it. How courses are placed
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