1. Sequences and Series

• Convergence of sequences of real numbers
• Comparison, root and ratio tests for convergence of series of real numbers

2. Differential Calculus

• Limits, continuity and differentiability of functions of one and two variables
• Rolle’s Theorem
• Mean value theorems
• Taylor’s theorem
• Indeterminate forms
• Maxima and minima of functions of one and two variables

3. Integral Calculus

• Fundamental theorems of integral calculus
• Double and triple integrals
• Applications of definite integrals
• Arc lengths, areas and volumes

4. Matrices

• Rank, inverse of a matrix
• Systems of linear equations
• Linear transformations
• Eigenvalues and eigenvectors
• Cayley‐Hamilton theorem
• Symmetric, skew‐symmetric and orthogonal matrices

5. Differential Equations

• Ordinary differential equations of the first order of the form y’ = f(x,y)
• Linear differential equations of the second order with constant coefficients

6. Descriptive Statistics and Probability

• Measure of Central tendency
• Measure of dispersion
• Skewness and Kurtosis
• Elementary analysis of data
• Axiomatic definition of probability and properties
• Conditional probability, multiplication rule
• Theorem of total probability
• Bayes’ theorem and independence of events

7. Random Variables

• Probability mass function
• Probability density function
• Cumulative distribution functions
• Distribution of a function of a random variable
• Mathematical expectation
• Moments and moment generating function
• Chebyshev’s inequality

8. Standard Distributions

• Binomial, negative binomial, geometric
• Poisson, hyper-geometric
• Uniform, exponential, gamma
• Beta and normal distributions
• Poisson and normal approximations of a binomial distribution
• Chi-square distribution
• t-distribution and F-distribution

9. Joint Distributions

• Joint, marginal and conditional distributions
• Distribution of functions of random variables
• Product moments, correlation
• Simple linear regression
• Independence of random variables

10. Limit Theorems

• Weak law of large numbers
• Central limit theorem (i.i.d. with finite variance case only)

11. Estimation

• Unbiasedness, consistency and efficiency of estimators
• Method of moments and method of maximum likelihood
• Sufficiency, factorization theorem
• Completeness
• Rao‐Blackwell and Lehmann‐Scheffe theorems
• Uniformly minimum variance unbiased estimators
• Rao‐Cramer inequality
• Confidence intervals for parameters of univariate normal, two independent normal, and one parameter exponential distributions

12. Testing of Hypotheses

• Basic concepts
• Applications of Neyman‐Pearson Lemma for testing simple and composite hypotheses
• Likelihood ratio tests

13. Sampling and Designs of Experiments

• Simple random sampling
• Stratified sampling
• Cluster sampling
• One-way, two-way analysis of variance
• CRD, RBD, LSD
• 2² and 2³ factorial experiments