MWT-01 |
Real Analysis |
3-Jan-22 |
Sequences and Series of real numbers: Sequences of real numbers, their convergnece, and limits. Cauchy sequences and their convergence. Monotonic sequences and their limits. Limits of standard sequences. Infinite series and its convergence of a series. comparison test, limit comparison test, D'Alembert's ratio test, Cauchy's nth root test, cauchy's condensation test and integral test. absolute convergence of series. Leibnitz's test for the convergence of alternating series. conditional convergence. convergence of power series and radius of convergence. |
MWT-02 |
Basic Probability and Random Variables |
6-Jan-22 |
Probabiliy: Random Experiments. Sample space and Algebra of Events (Event space). Relative frequency and Axiomatic definitons of probability. Properties of probability function. Addition theorem of probability fucntion (inclusion exclusion principle). Geometric probability. Boole's and Bonferroni's inequaliites. conditional probability and multiplication rule. theorem of total probability and Bayes' theorem. pairwise and mutual independence of events. Univariate Distributions: Definition of random variabels. cumulative distribution fucntion (cdf) of a random variable. Discrete and Continuous random variables. probability mass fucntion (pmf) and probability density fucnion (pdf) of a random variabel. Distribution (cdf, pmf, pdf) of a function of a random variable using transformation of variable and Jacobian method. mathematical expectation and moments. mean, median, mode, variance, standard deviation, Coefficient of variation, quantiles , quartiles, coefficient of variation, and measures of Skewness and Kurtosis of a probability distribution. moment generating function (mgf), its properties and uniqueness, markov and Chebyshev inequalities and their applications. |
MWT-03 |
Standard Distribution & Limit theorams |
9-Jan-22 |
Standard Univariate Distribution: Degenerate, Bernoulli, Binomial, Negative binomial, Geometric, Poisson, Hypergeometric, Uniform, Exponential, Double exponential, Gamma, Beta( of first and second type), Normal and Cauchy distributions, their properties, intrrelations, and limiting (approximation) cases. Limit Theorem: Convergence in probability, convergence in distribution and their inter relations, weak law of large numbers and Central Limit Theorem (i.i.d. case) and their applications. |
MWT-04 |
Linear Algebra |
12-Jan-22 |
Matrices and Determinants: Vector spaces with real field. Subspaces and sum of subspaces. Span of a set. Linear dependence and independnece. Dimension and basis. Algebra of matrices. Standard matrices (symmetric and Skew symmetric matrices, Hermitian and skew hermitian matrices, Orthogonal and Unitary matrices, indempotent and Nilpotent matrices) . Definition, properties and applications of determinants. singular matrices and their properties. trace of a matrix. Adjoint and inverse of a matrixd and related properties. Rank of a matrix, row -rank, column-rank, standard thorems on ranks, rank of the sum and the product of two matrices. Row reduction and echelon forms. Partitioning of matrices and simple properties. consistent and inconsistent system of linear equations. properties of solutions of system of linear equations. use of determinants in solution to the system of linear equations. Cramer's rule. Characteristic roots and Characteristic vectors. properties of charcterisitc roots and vectors. Cayley Hamiton theorem. |
MWT-05 |
Joint Distributions & Sampling Distribution |
15-Jan-22 |
Multivariate Distributions: Definition of random vectors. Joint and marginal c.d.f.s of a random vector. Discrete and continuous type random vectors. Joint and marginal p.m.f., joint and marginal pdf. Conditional cdf, conditional pmf and conditional pdf. Independence of random variables. Distribution of fucntions of random vectors using transformation of variables and Jacobian method. mathematical expectation of functions of random vectors. Joint moments, Covariance and Correlation. Joint moment generating function and its properties. Uniqueness of joint mgf and its applications. Conditional moments, conditional expectations and conditional variance. Additive properties of Binomial, Poisson, Negative Binomial, Gamma and Normal Distribution using their mgf. Standard Multivariate Distributions: Multinomial distribution as a generalization of binomial distribtion and its properties (moments, correlation, marginal distribution, additive property). Bivariate normal distribution, its marginal and conditional distributions and related properties. Sampling Distribution : Definitions of random sample, parameter and statistic. sampling distribution of a statistic. Order Statistics: Definition and distributionof the rth order statisc (d.f. and p.d.f for i.i.d. case for continuous distributions). Distribution (c.d.f., pmf., pdf) of smallest and largest order statistics (i.i.d. case for discrete as well as continuous distributions). Central Chi-square distribution : Definition and derivation of pdf of central l2 distribution. central student's t-distribution: Definition and derivation of pdf of central student's t-distribution with n d.f. Properties and limiting form of central t-distribution Sneecor's central F-distribution: Definition and derivation of pdf of Snedecor's central F-distribution with (m,n) d.f.. Properties of Central F-distribution, distribution of the reciprocal of F-distribution. relationship between t, F and l2 distributions. |
MWT-06 |
Estimation |
18-Jan-22 |
Estimation: Unbiasedness. Sufficiency of a statistic. Factorization theorem. Complete statistic. Consistency and relative efficiency of estimators. Uniformly Minimum variance unbiased estimator (UMVUE). Rao-Blackwell and Lehmann-Scheffe theorems and their applications. Cramer-Rao inequality and UMVUEs. Methods of Estimation: Method of moments, method of maximum likelihood, invariance of maximum likelihood estimators. |
MWT-07 |
Integral Calculus |
21-Jan-22 |
Integral Calculus: Fundamental theorems of integral calculus (single integral). Lebnitz's rule and its applications. Differentiation under integral sign. Improper integrals. Beta and Gamma integrals: properties and relationship between them. Double integrals. Change of order of integration. Transformation of variabels. Applications of definite integrals. Arc lenghts, areas and Volumes. |
MWT-08 |
Testing of Hytothesis |
24-Jan-22 |
Testing of Hypotheses: Null and alternative hypotheses (simple and composite), Type-I & TypeII errors. Critical region. Level of significance, size and power of a test, p-value. Most powerful critical regions and most powerful (MP) tests. Uniformly most powerful (UMP) tests. Neyman Pearson Lemma (without proof) and its applications to construction of MP and UMP tests for paramter of single parameter parametric families. likelihood ratio tests for parameters of univariate normal distribution. Least squares estimation and its applications in simple linear regression models. Confidence intervals and confidence coefficient. confidence intervlas for the parameters of univariate normal, two independnet normal, and exponential distribution. |
FLT-01 |
Full Length Test |
27-Jan-22 |
As per Exam Pattern |
FLT-02 |
Full Length Test |
31-Jan-22 |
As per Exam Pattern |
FLT-03 |
Full Length Test |
4-Feb-22 |
As per Exam Pattern |
FLT-04 |
Full Length Test |
8-Feb-22 |
As per Exam Pattern |