STAT 7032 Probability Spring 2020

Take-home Final

Tentative Schedule based on [D] with Homework

Read the material before it is covered in class!

1. (Jan 13): Sect. 1.1, 1.2, 1.3:
2. (Jan 20 - no class Monday:) Sect. 1.3, 1.4:
3. (Jan 27): Sect. 1.4, 1.5
   Hwk3 is due Mo Feb 3
4. (Feb 3): Sect. 1.5/1.6, 1.7,
   Hwk4 is due We Feb 12
5. (Feb 10): Sect. 1.7, 2.1.1, 2.1.2 2.1.3 No class Feb 10
   Hwk5 due Mo, Feb 17
6. (Feb 17): Sect. 2.1.4 2.2, 2.3
   Hwk 6 (not to be turned in)
7. (Feb 24): Sect. 2.3, Exam 1 Fri Feb 28 3-5pm CGD215
   Prep Questions
8. (March 2): Sect. 2.4
   Hwk 7 is due Monday, March 9
9. (March 9): Sect 2.5, 3.2
   Hwk 8 is due when classes resume
10. (March 16): Spring break

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    Revised for virtual mode

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11. March 27: Sect 3.2 (continued) online using 2019-Notes Ch 9+10

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    Homework 9: Exercises 9.2, 9.7 9.12 from 2019-Notes Ch 9+10 due Tue, March 31 as file upload on Canvas not BB this time.

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12. (March 30): Sect. 3.3 Characteristic functions Homework 10: due Monday, April 6 by email is not assigned.
13. (Apr 6): Review 1+2+3, and Exam 2,
14. (Apr 13): Sect. 3.4 Central Limit Theorem
    Hwk 11 : Turn in solutions for two Exercises of your choice from Exercises 11.1-11.7 in the notes. Due Monday, April 20, 2020.
15. (Apr 20): Sect. 3.10 Multivariate normal distribution and mutlivariate CLT, Review
16. (Apr 27): Final Exam: Monday, April 27, 9:45–11:45 a.m. online.

Brief Description of the course:

Textbooks

• Primary: Probability: Theory and Examples, Edition 5 by Rick Durrett. A PDF is available from the website of the author.
  
• Secondary: [B, G, PS, V, R] listed below.

Exams

• Exam 1: Fri Feb 28 3-5pm CGD215
• Exam 2: Fri, April 10 3-5pm (via Webex)
• Final Exam: Monday, April 27, 9:45–11:45 a.m. 60WCHARL 250 Take home due Sat, May 2 NOON.

Grading

Important theorems

• Borel-Cantelli Lemmas [D, Thms 2.3.1, 2.3.6], [B,Thms 4.3, 4.4], [G,Ch Thms 2.18.1, 2.18.2], [R,Propositions 4.5.1, 4.5.2], [PS,?], [V,?]
• Kolmogorov's Zero-One law [D, Thm 2.5.3], [B,Thms 4.5, 22.3], [G,Thms 1.5.1, 5.10.6], [R,Thm 4.5.3], [PS,?], [V,?]
• Kolmogorov's maximal inequality for sums of independent random variables [D, Thm 2.5.5], [B,Thm 22.4], [G,Thm 3.1.6], [PS,?], [V,?] Other maximal inequalities: Sokorohod's [R,Proposition 7.3.1], Etemadi's [B,Theorem 22.5]
• Convergence of series:
  • [D, Thm. 2.5.6], [B,Theorem 22.6], [R,Theorem 7.3.3], [PS,?], [V,?]
  • Kolmogorov's three series theorem [D, Thm 2.5.8], [B,Theorem 22.8], [R,Theorem 7.61.], [PS,?], [V,?]
• Law of large numbers for independent random variables [D, Thms 2.4.1, 2.5.10], [B,Thm 22.1], [G,Thm 6.6.1], [R,Theorem 7.5.1]
• Weak convergence:
  1. Portmanteau theorem:[D,Thm 3.2.9], [B,Theorem 25.8], [R,Theorem 8.4.1]
  2. Skorohod Representation Theorem ($\mathbb{R}^1$ - version only). [D, Thm 3.2.8], [B, Thm 25.6] [G, Thm 5.13.1], [R, Theorem 8.3.2]
  3. Sheffe's Lemma [B,Theorem 16.12], [R,Lemma 8.2.1].
  4. Slutsky's theorem [D, Exercise 3.2.13], [R,Theorem 8.6.1]
  5. convergence of types [B,Theorem 14.2], [R,Theorem 8.7.1]
• The Central Limit Theorem
  1. for the sums of i.i.d random variables [D, Thm 3.4.1], [B,Thm 27.1] [G,Thm 7.1.1], [R,Theorem 9.7.1]
  2. Lindeberg's Central Limit Theorem for the sums of independent random variables [D, Thm 3.4.10], [B,Thm 27.2], [G,7.2.1], [R,Theorem 9.8.1], [PS,?], [V,?]
  3. Lyapunov's theorem [D, Exercise 3.4.12], [B,Theorem 27.2], [R,Corollary 9.8.1], [PS,?], [V,?]
• The δ method: [D,?], [B,Exercise 27.10], [R], [PS,?], [V,?]

References

[B]

P. Billingsley, Probability and Measure IIIrd edition

[D]

R. Durrett, Probability: Theory and Examples, Edition 5.1 (online)

[G]

A. Gut, Probability: a graduate course

[R]

S. Resnik, A Probability Path, Birkhause 1998

[PS]

S M. Proschan and P. Shaw, Essential of Probability Theory for Statistitcians, CRC Press 2016

[V]

S.R.S. Varadhan, Probability Theory, (online pdf from 2000)