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Syllabus ( MATH 581 )


   Basic information
Course title: Probability And Mathematical Statistics I
Course code: MATH 581
Lecturer: Prof. Dr. Nuri ÇELİK
ECTS credits: 7.5
GTU credits: 3 (3+0+0)
Year, Semester: 1/2, Fall and Spring
Level of course: Second Cycle (Master's)
Type of course: Area Elective
Language of instruction: English
Mode of delivery: Face to face
Pre- and co-requisites: None
Professional practice: No
Purpose of the course: The main goal of this course is to introduce the students to advence probability theory and mathematical statistics and encourage them to do research on paremeter estimation and hypothesis testing
   Learning outcomes Up

Upon successful completion of this course, students will be able to:

  1. explain fundamental concept of the advance probability theory and mathematical statistics.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results
    4. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
  2. demostrate how to construct the extension and check the uniqness.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
  3. explain the concept of Measures; General measures, Outher measures, Maesure in Euclidean Space.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
  4. explain the properties of integral and integral with respect to Lebesgue measure.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
    2. Homework assignment
  5. demostrate the Product Measure and Fubini Theorem.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
    2. Homework assignment
  6. demostrate the characterization of Exponential distribution, and relate to poisson process

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Homework assignment
  7. percieve the principles of data reduction and relate this to point estimation

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Homework assignment
   Contents Up
Week 1: Probability Measures; spaces,claceses of sets,probability measure, Lebesgue measure on the unit interval.
Week 2: Existence and extension; Construction of the extension, Unikness and pi-lambda theorem; monotone classes.
Week 3: Simple Random Variables; Independence, existence of independence sequences, expected values, inequalities.
Week 4: Law of large numbers; Strong law, Weak law.
Week 5: Markow Chains; Definitions-Higher-Order transitions; Theorem-Transience and Persistence
Week 6: Measures; General measures, Outher measures, Maesure in Euclidean Space
Week 7: Measurable functions and Mappings; Distribution functions.
Week 8: Integration; Integral, Properties of integral; Integral with respect to Lebesgue measure.
Week 9: Product Measure and Fubini Theorem.
Week 10: Midterm exam
Week 11: Random Variable and Distributions; Expected Values.
Week 12: Sum of Independent random variables; Strong Law of large numbers, Weak law, Characteristic function, Kolmogorow zero-one law.
Week 13: Poisson process; Characterization of Exponential distribution,
Week 14: Principles of data reduction; sufficient statistics, minimal sufficient statistics, complete sufficient statistics, Likelihood function.
Week 15*: Point estimation; method of finding estimators; maximum likelihood estimators, invariant estimators, mean square error, sufficiency and unbiasedness, consistency.
Week 16*: Final exam
Textbooks and materials: Probability and Measure, by Patrick Billingsley
Statistical Inference, by Casella and Berger,
Recommended readings: [2] Arnold, S. F. Mathematical Statistics, Prentice Hall
[3] Introduction to Mathematical Statistics by Hogg and Craig
[4] Introduction to Probability Models by Ross
  * Between 15th and 16th weeks is there a free week for students to prepare for final exam.
Assessment Up
Method of assessment Week number Weight (%)
Mid-terms: 10 30
Other in-term studies: 0
Project: 0
Homework: 1,2,3,4,5,6,7 20
Quiz: 0
Final exam: 16 50
  Total weight:
(%)
   Workload Up
Activity Duration (Hours per week) Total number of weeks Total hours in term
Courses (Face-to-face teaching): 3 14
Own studies outside class: 5 14
Practice, Recitation: 0 0
Homework: 5 7
Term project: 0 0
Term project presentation: 0 0
Quiz: 0 0
Own study for mid-term exam: 15 1
Mid-term: 3 1
Personal studies for final exam: 15 1
Final exam: 3 1
    Total workload:
    Total ECTS credits:
*
  * ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
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