15-02-2013, 09:28 AM
RANDOM PROCESS
[attachment=50973]
Syllabus :
Unit 1: Probability and Random Variable
Definition, sample space, conditional probability, Baye’s theorem, Bernouli’s trials, Asymptotic theorems, Poison’s theorem and random points;
Random Variable- Definition, Continuous and discrete random variable, distribution and density functions; Conditional distribution; One random variable- Mean, variance, moments, characteristic functions; Two random variables- Mean, variance, moments, characteristic functions; Moments and conditional statistics; Transformation of random variables; Random process; Mean, Correlation and Covariance; Stationarity; transmission of a random process through a linear filter, power spectral density, Gaussian process;
Unit 2: Stochastic Process
Definition, first and second order statistics, Mean, Correlation and Covariance; Ergodic process; Spectral Representation of Stochastic process; Random walk, Brownian motion, Thermal noise, Poisson point, Shot noise, Modulation, Cyclostationary Process, Band limited Process;
Unit 3: Estimation
Spectral Estimation, Extrapolation and system identification, mean square estimation, prediction, filtering and prediction; Kalman Filters;
UNCERTAINITY
Uncertainity, chance, element of luck still plays important role in our lives, science, technology.
Example : Success of space mission, launching of satellite, noise in signal, outcome of a surgical operation, failure of an instrument /tool, accuracy of a chemical / medical test, reaction of medicine, death, injury, disease, false imprisonment, wrongful execution, judgment regarding suitability of a person for a job, Stock exchange index, weather condition, who will win a game, timely arrival, time required in queue in bank, petrol pump etc etc.
It is important and CRUCIAL to get things right.
We have to take decisions.
Impact of uncertainty is to be eliminated / minimised
Random Process :
This subject deals with uncertainty.
Based on data and information about past happenings of similar events
Study, analysis, modeling, inferences
To help take decisions
So that impact of uncertainty is eliminated/minimised.
Will it rain to-day ? Will you take umbrella with you ?
What will you do to reach exam hall in time ?
Which doctor / hospital / diagnostic centre to approach ?
Which subject / branch / optional to study and in which institution ?
Who will win the final ?
Which brand/company will you choose to buy the equipment from ?
How to hire the most suitable person for a job.
Probability :
Deals with populations or aggregates of individuals rather than individual.
Deals with variation, numerically specified data,
Deals with population that occur in nature and are subject to a large number of variations,
The logic used is is inductive ( mathematics uses deductive logic),
Inferences are uncertain,
Gives idea about some characteristics of a group,
May be wrong if applied to an individual,
Results may lead to fallacious conclusions if they are quoted shorn off their context or manipulated
Probability :Few applications
Composition analysis of rocks : Homogeneous samples to several laboratories, Data compilation, Certified values estimated.
Quality assurance: High degree of uniformity of products of a plant.
Elimination of variability between products.
Quality control: Action taken to minimise variation between two individual products. Ensuring product characteristics within specified tolerence limit.
Acceptance sampling of lot of shipped goods.
Reliability : Specified in terms of probability. Reliability of an item is probability that it will function satisfactorily within specified limit for atleast a specified period of time under specified condition.
Life testing: method ofestimating reliability. Samples are randomly taken from lot. Put on test. Times of failure is observed.
Signal processing etc.
[attachment=50973]
Syllabus :
Unit 1: Probability and Random Variable
Definition, sample space, conditional probability, Baye’s theorem, Bernouli’s trials, Asymptotic theorems, Poison’s theorem and random points;
Random Variable- Definition, Continuous and discrete random variable, distribution and density functions; Conditional distribution; One random variable- Mean, variance, moments, characteristic functions; Two random variables- Mean, variance, moments, characteristic functions; Moments and conditional statistics; Transformation of random variables; Random process; Mean, Correlation and Covariance; Stationarity; transmission of a random process through a linear filter, power spectral density, Gaussian process;
Unit 2: Stochastic Process
Definition, first and second order statistics, Mean, Correlation and Covariance; Ergodic process; Spectral Representation of Stochastic process; Random walk, Brownian motion, Thermal noise, Poisson point, Shot noise, Modulation, Cyclostationary Process, Band limited Process;
Unit 3: Estimation
Spectral Estimation, Extrapolation and system identification, mean square estimation, prediction, filtering and prediction; Kalman Filters;
UNCERTAINITY
Uncertainity, chance, element of luck still plays important role in our lives, science, technology.
Example : Success of space mission, launching of satellite, noise in signal, outcome of a surgical operation, failure of an instrument /tool, accuracy of a chemical / medical test, reaction of medicine, death, injury, disease, false imprisonment, wrongful execution, judgment regarding suitability of a person for a job, Stock exchange index, weather condition, who will win a game, timely arrival, time required in queue in bank, petrol pump etc etc.
It is important and CRUCIAL to get things right.
We have to take decisions.
Impact of uncertainty is to be eliminated / minimised
Random Process :
This subject deals with uncertainty.
Based on data and information about past happenings of similar events
Study, analysis, modeling, inferences
To help take decisions
So that impact of uncertainty is eliminated/minimised.
Will it rain to-day ? Will you take umbrella with you ?
What will you do to reach exam hall in time ?
Which doctor / hospital / diagnostic centre to approach ?
Which subject / branch / optional to study and in which institution ?
Who will win the final ?
Which brand/company will you choose to buy the equipment from ?
How to hire the most suitable person for a job.
Probability :
Deals with populations or aggregates of individuals rather than individual.
Deals with variation, numerically specified data,
Deals with population that occur in nature and are subject to a large number of variations,
The logic used is is inductive ( mathematics uses deductive logic),
Inferences are uncertain,
Gives idea about some characteristics of a group,
May be wrong if applied to an individual,
Results may lead to fallacious conclusions if they are quoted shorn off their context or manipulated
Probability :Few applications
Composition analysis of rocks : Homogeneous samples to several laboratories, Data compilation, Certified values estimated.
Quality assurance: High degree of uniformity of products of a plant.
Elimination of variability between products.
Quality control: Action taken to minimise variation between two individual products. Ensuring product characteristics within specified tolerence limit.
Acceptance sampling of lot of shipped goods.
Reliability : Specified in terms of probability. Reliability of an item is probability that it will function satisfactorily within specified limit for atleast a specified period of time under specified condition.
Life testing: method ofestimating reliability. Samples are randomly taken from lot. Put on test. Times of failure is observed.
Signal processing etc.