MA3303 Probability and complex function

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UNIT -1
1.Exponential Distribution( mean, variance)**
2. Poisson distribution
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3. binomial ,geometric distribution
rare 
4. Bayes theorem
UNIT-2
1. Correlative and regression**
2. Covariance
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3. central limit theorem 
UNIT-3
1.Cartesian and Harmonic function**
2.Bilinear transformation**
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Unit 4
1.Cauchy's residue theorem 

2.Countour integration **
3.Laurent series 
UNIT-5
1.Convolution theorem (8)
2.f(t)=?(8)
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2.22 factorial design
UNIT-5
1.CMethod of variation of parameters**
2.Homogenous equation of Euler’s **and Legendre’s type
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 *These questions are expected for the exams This may or may not be asked for exams .

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Syllabus

UNIT I PROBABILITY AND RANDOM VARIABLES
Axioms of probability – Conditional probability – Baye’s theorem - Discrete and continuous random variables – Moments – Moment generating functions – Binomial, Poisson, Geometric, Uniform, Exponential and Normal distributions – Functions of a random variable.
UNIT II TWO-DIMENSIONAL RANDOM VARIABLES
Joint distributions – Marginal and conditional distributions – Covariance – Correlation and linear regression – Transformation of random variables – Central limit theorem (for independent and identically distributed random variables).
UNIT III ANALYTIC FUNCTIONS
Analytic functions – Necessary and sufficient conditions for analyticity in Cartesian and polar coordinates - Properties – Harmonic conjugates – Construction of analytic function - Conformal mapping – Mapping by functions 1 2 z w  z  c, cz, , - Bilinear transformation.
UNIT IV COMPLEX INTEGRATION
Line integral - Cauchy’s integral theorem – Cauchy’s integral formula – Taylor’s and Laurent’s series – Singularities – Residues – Residue theorem – Application of residue theorem for evaluation of real integrals – Applications of circular contour and semicircular contour (with poles NOT on real axis).
UNIT V ORDINARY DIFFERENTIAL EQUATIONS
Higher order linear differential equations with constant coefficients - Method of variation of parameters– Homogenous equation of Euler’s and Legendre’s type – System of simultaneous linear first order differential equations with constant coefficients - Method of undetermined coefficients

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