DIPS ACADEMY has truly been one of the most profound learning experience where I gained the confidence to move ahead in my carrer with the opportunity to put myself on the forefront in dealing mathematics. DIPS programme is very well designed with very helpful test series. Dubey Sir and other faculty are superb. The lectures of Dubey Sir are the most intersting and inspiring.
I am very contented with the dips family specially dubey sir, he's an excellent mentor with amazing teaching skills.The mathematical atmosphere along with one to one interaction, dips material and doubt clearing sessions provided by DIPS ACADEMY were very helpful throughout the journey.
" The guidance of Dubey Sir, a teacher with immense knowledge and wonderful teaching skills, was of great help for me. His approach towards tackling problems, in particular, is pecular and worth emulating I thank him for his invaluable guidance and continuous motivation. "
I'm Mamta Kumari and I secured AIR 47 in IIT - JAM Mathematics. I am very thankful to Dubey Sir, Amit Sir and all the other faculty members for their teachings and guidance. The classes used to be very interactive and environment was very positive for learning. I have learned a lot from Dips academy and I'm very thankful to all the faculties for teaching us the actual meaning of Mathematics in the best possible way.
Real Analysis
Sequences and Series of Real Numbers: Convergence of sequences, bounded and monotone
sequences, Cauchy sequences, Bolzano-Weierstrass theorem, absolute convergence, tests of
convergence for series – comparison test, ratio test, root test; Power series (of one real variable), radius
and interval of convergence, term-wise differentiation and integration of power series.
Functions of One Real Variable: L imit, continuity, intermediate value property, differentiation, Rolle’s
Theorem, mean value theorem, L'Hospital rule, Taylor's theorem, Taylor’s series, maxima and minima,
Riemann integration (definite integrals and their properties), fundamental theorem of calculus.
Multivariable Calculus and Differential Equations
Functions of Two or Three Real Variables: : Limit, continuity, partial derivatives, total derivative, maxima
and minima.
Integral Calculus: Double and triple integrals, change of order of integration, calculating surface areas
and volumes using double integrals, calculating volumes using triple integrals.
Differential Equations: Bernoulli’s equation, exact differential equations, integrating factors, orthogonal
trajectories, homogeneous differential equations, method of separation of variables, linear differential
equations of second order with constant coefficients, method of variation of parameters, Cauchy-Euler
equation.
Linear Algebra and Algebra:
Matrices: Systems of linear equations, rank, nullity, rank-nullity theorem, inverse, determinant,
eigenvalues, eigenvectors.
Finite Dimensional Vector Spaces: Linear independence of vectors, basis, dimension, linear
transformations, matrix representation, range space, null space, rank-nullity theorem.
Groups: Cyclic groups, abelian groups, non-abelian groups, permutation groups, normal subgroups,
quotient groups, Lagrange's theorem for finite groups, group homomorphisms.
The Mathematical Statistics (MS) Test Paper comprises following topics of Mathematics (about 30% weight) and Statistics (about 70% weight).
Mathematics
Sequences and Series of real numbers: Sequences of real numbers, their convergence, and limits.
Cauchy sequences and their convergence. Monotonic sequences and their limits. Limits of standard
sequences. Infinite series and its convergence, and divergence. Convergence of series with non-negative
terms. Tests for convergence and divergence of a series. Comparison test, limit comparison test,
D’Alembert’s ratio test, Cauchy’s 𝑛
𝑡ℎ
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.
Differential Calculus of one and two real variables: Limits of functions of one real variable. Continuity and
differentiability of functions of one real variable. Properties of continuous and differentiable functions of one
real variable. Rolle's theorem and Lagrange's mean value theorems. Higher order derivatives, Lebnitz's rule
and its applications. Taylor's theorem with Lagrange's and Cauchy's form of remainders. Taylor's and
Maclaurin's series of standard functions. Indeterminate forms and L' Hospital's rule. Maxima and minima of
functions of one real variable, critical points, local maxima and minima, global maxima and minima, and point
of inflection. Limits of functions of two real variables. Continuity and differentiability of functions of two real
variables. Properties of continuous and differentiable functions of two real variables. Partial differentiation
and total differentiation. Lebnitz's rule for successive differentiation. Maxima and minima of functions of two
real variables. Critical points, Hessian matrix, and saddle points. Constrained optimization techniques (with
Lagrange multiplier).
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 variables.
Applications of definite integrals. Arc lengths, areas and volumes.
Matrices and Determinants: Vector spaces with real field. Subspaces and sum of subspaces. Span of a
set. Linear dependence and independence. Dimension and basis. Algebra of matrices. Standard matrices
(Symmetric and Skew Symmetric matrices, Hermitian and Skew Hermitian matrices, Orthogonal and Unitary
matrices, Idempotent and Nilpotent matrices). Definition, properties and applications of determinants.
Evaluation of determinants using transformations. Determinant of product of matrices. Singular and nonsingular matrices and their properties. Trace of a matrix. Adjoint and inverse of a matrix and related
properties. Rank of a matrix, row-rank, column-rank, standard theorems 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 characteristic roots and vectors. Cayley Hamilton theorem.
Statistics
Probability: Random Experiments. Sample Space and Algebra of Events (Event space). Relative frequency
and Axiomatic definitions of probability. Properties of probability function. Addition theorem of probability
function (inclusion exclusion principle). Geometric probability. Boole's and Bonferroni's inequalities.
Conditional probability and Multiplication rule. Theorem of total probability and Bayes’ theorem. Pairwise
and mutual independence of events.
Univariate Distributions:Definition of random variables. Cumulative distribution function (c.d.f.) of a random
variable. Discrete and Continuous random variables. Probability mass function (p.m.f.) and Probability density
function (p.d.f.) of a random variable. Distribution (c.d.f., p.m.f., p.d.f.) 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 (m.g.f.),
its properties and uniqueness. Markov and Chebyshev inequalities and their applications.
Standard Univariate Distributions: 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, interrelations, and limiting (approximation) cases.
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 p.d.f.. Conditional
c.d.f., conditional p.m.f. and conditional p.d.f.. Independence of random variables. Distribution of functions 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 m.g.f. and its applications. Conditional moments, conditional
expectations and conditional variance. Additive properties of Binomial, Poisson, Negative Binomial, Gamma
and Normal Distributions using their m.g.f..
Standard Multivariate Distributions: Multinomial distribution as a generalization of binomial distribution and
its properties (moments, correlation, marginal distributions, additive property). Bivariate normal distribution,
its marginal and conditional distributions and related properties.
Limit Theorems: 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.
Sampling Distributions: Definitions of random sample, parameter and statistic. Sampling distribution of a
statistic. Order Statistics: Definition and distribution of the 𝑟
𝑡ℎ order statistic (d.f. and p.d.f. for i.i.d. case for
continuous distributions). Distribution (c.d.f., p.m.f., p.d.f.) 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 p.d.f. of central 𝜒2 distribution with 𝑛 degrees of freedom (d.f.) using m.g.f.. Properties of central 𝜒2
distribution, additive property and limiting form of central 𝜒2 distribution. Central Student's 𝒕-distribution:
Definition and derivation of p.d.f. of Central Student's 𝑡-distribution with 𝑛 d.f., Properties and limiting form of
central 𝑡-distribution. Snedecor's Central 𝑭-distribution: Definition and derivation of p.d.f. 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 𝜒2 distributions.
Estimation: Unbiasedness. Sufficiency of a statistic. Factorization theorem. Complete statistic. Consistency
and relative efficiency of estimators. Uniformly Minimum variance unbiased estimator (UMVUE). RaoBlackwell 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. Least squares estimation and its applications in simple linear regression models.
Confidence intervals and confidence coefficient. Confidence intervals for the parameters of univariate normal,
two independent normal, and exponential distributions.
Testing of Hypotheses: Null and alternative hypotheses (simple and composite), Type-I and Type-II
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 parameter of single parameter
parametric families. Likelihood ratio tests for parameters of univariate normal distribution.
DIPS ACADEMY has truly been one of the most profound learning experience where I gained the confidence to move ahead in my carrer with the opportunity to put myself on the forefront in dealing mathematics. DIPS programme is very well designed with very helpful test series. Dubey Sir and other faculty are superb. The lectures of Dubey Sir are the most intersting and inspiring.
I am very contented with the dips family specially dubey sir, he's an excellent mentor with amazing teaching skills.The mathematical atmosphere along with one to one interaction, dips material and doubt clearing sessions provided by DIPS ACADEMY were very helpful throughout the journey.
" The guidance of Dubey Sir, a teacher with immense knowledge and wonderful teaching skills, was of great help for me. His approach towards tackling problems, in particular, is pecular and worth emulating I thank him for his invaluable guidance and continuous motivation. "
I'm Mamta Kumari and I secured AIR 47 in IIT - JAM Mathematics. I am very thankful to Dubey Sir, Amit Sir and all the other faculty members for their teachings and guidance. The classes used to be very interactive and environment was very positive for learning. I have learned a lot from Dips academy and I'm very thankful to all the faculties for teaching us the actual meaning of Mathematics in the best possible way.