NOC:Modeling Stochastic Phenomena for Engineering Applications: Part-1

Lecture 1 - Stirling's Approximation

Lecture 2 - Fourier Transforms and characteristic function

Lecture 3 - Dirac Delta function

Lecture 4 - Applications of delta function and Generating functions

Lecture 5 - Laplace Transforms and Convolution theorem

Lecture 6 - Generating function for discrete variables and Binomial distribution

Lecture 7 - Bernoulli and Poisson distributions

Lecture 8 - Waiting time distributions; Gaussian approximation to Poisson distribution

Lecture 9 - Introduction to Central Limit Theorem

Lecture 10 - Proof of Central Limit Theorem (CLT)

Lecture 11 - Universality of Normal distribution and Exceptions

Lecture 12 - Introduction to Random Walk: Extension of Central Limit Theorem

Lecture 13 - Random walk and Diffusion coefficient: Conditional and Transition

Lecture 14 - Characteristics of Stochastic Phenomena: Markov Processes

Lecture 15 - Propagating Markov processes via Transition Probability Matrix with

Lecture 16 - Chapman-Kolmogorov Equation for Multistep Transition probability and solution

Lecture 17 - Transient solutions and Continuous time Markov process

Lecture 18 - Exact solution to Symmetric (or unbiased) one-dimensional Random walk (1-D RW)

Lecture 19 - Properties of the solution for 1-D unbiased RW

Lecture 20 - 1-D unbiased RW: Asymptotic form of occupancy probability and transition

Lecture 21 - Solution to the problem of 1-D Random Walk with Bias

Lecture 22 - Generalized Random Walk with Bias and Pausing

Lecture 23 - Effect of Pausing on Mean and Variance of Random walk

Lecture 24 - Random-walk in the presence of reflecting barrier

Lecture 25 - Boundary conditions for reflected Random-Walk and formulating absorbing

Lecture 26 - The survival probability and first-passage time distribution for Random walker

Lecture 27 - Random Walk with Bias and Absorber

Lecture 28 - Drift and Survival probability for Random walk with bias and absorber

Lecture 29 - Introduction to gambler's ruin problem

Lecture 30 - Solution for ultimate winning probability in Gambler's ruin problem

Lecture 31 - Solution to gambler's ruin problem with site dependent jump probabilities

Lecture 32 - Fourier transform method of solving lattice Random walks

Lecture 33 - Two and higher dimensional Random walks

Lecture 34 - Formulating the problem of Probability of Return to the origin

Lecture 35 - Relationship between occupancy probability and first-time-return probability

Lecture 36 - Proof of Polya’s theorem on the probability of return

Lecture 37 - Return probability estimates in various dimensions and effect of bias in 1-D

Lecture 38 - Dependence of first time return probability (Fk) on steps

Lecture 39 - Equilibrium solutions in lattice random walk models

Lecture 40 - Equilibrium solution to Ehrenfest's flea model

Lecture 41 - Differential equation formulation of stochastic phenomena

Lecture 42 - Derivation of Fokker-Planck equation

Lecture 43 - Generalized transition probability functions for Fokker-Planck equation

Lecture 44 - Solution to 1-D Fokker-Planck equation for free particle: Method of Fourier

Lecture 45 - General non-gaussian solution to translationally invariant Chapman-Kolmogorov

Lecture 46 - Cauchy distribution, power-law and other non-gaussian solutions

Lecture 47 - Wiener process and solution to absorbing barrier problems from Fokker-Planck

Lecture 48 - Application of Fourier Sine transform for single absorber problem

Lecture 49 - Setting up Langevin equation for velocity fluctuations of Brownian particles

Lecture 50 - Understanding the origin of systematic and random parts of force from kinetic

Lecture 51 - Kinetic derivation of a formula for delta-correlated random force

Lecture 52 - Mean square velocity, thermal equilibrium and relationship between relaxation

Lecture 53 - Velocity autocorrelation in Brownian motion

Lecture 54 - Derivation of Stokes-Einstein relationship between diffusion coefficient and

Lecture 55 - Alternative derivation of Stokes-Einstein relationship and Brownian motion with

Lecture 56 - Numerical simulation of the Langevin equation

Lecture 57 - Derivation of Klein-Kramers equation from Langevin equation for joint

Lecture 58 - Illustrative solutions to the Klein-Kramers equation

Lecture 59 - Numerical simulation: Sampling from general distributions and Central

Lecture 60 - Numerical simulation of Random walk trajectories and method of solving Fokker