Syllabus
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Course Code: ST-101 Course Name: Measure and Probability Theory |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Fields; sigma field, sigma-field generated by a class of subsets, Borel fields. Sequence of sets, limsup and liminf of sequence of sets, random variables, distribution function. Measure, probability measure, properties of a measure, Concept of outer measures, inner measures, lebesgue measures and Lebesgue-Stieltjes measure. |
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| 2 | Measurable functions: Sequence and algebra of measurable functions. convergence in measure. Integration of measurable function. Bounded convergence theorem, Fatou's Lemma, Monotone convergence theorem, General lebesgue integral, Dominated convergence theorem. |
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| 3 | Borel-Contelli Lemma, Borel 0-1 law, Kolmogorov's 0-1 law, Tchebycheff's and Kolmogorov's inequalities, various modes of convergence: in probability, almost sure, in distribution and in mean square and their interrelationship. |
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| 4 | Laws of large numbers for i.i.d. Sequences. Characteristic function its uniqueness, continuity and inversion formula. Applications of characteristic functions. Central limit theorems: De Moivre’s-Laplace, Liapounov, Lindeberg-Levy and their applications |
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