# Statistics 100a Homework #1 Solutions

**Course: MATH 170A, Probability Theory, Lecture 2, Spring 2017****Prerequisite:** Math 32B and Math 33A. Not open to students with credit for Electrical Engineering 131A or Statistics 100A.**Course Content:** Probability distributions, random variables and vectors, expectation. *Last update:* 14 June 2017**Instructor:** Steven Heilman, heilman(@-symbol)ucla.edu**Office Hours:** Fridays, 9AM-11AM, MS 5634**Lecture Meeting Time/Location:** Monday, Wednesday and Friday, 1PM-150PM, MS 5127**TA:** Tianqi Wu, timwu(@-symbol)ucla.edu**TA Office Hours:** Thursdays 3PM-5PM, MS 6146**Discussion Session Meeting Time/Location:** Tuesday, 1PM-150PM, MS 5127**Required Textbook:** D. P. Bertsekas and John N. Tsitsiklis, Introduction to Probability, 2nd edition. (The book is freely available online, **though some sections are ordered differently than the textbook**.)**Other Textbooks (not required):**__Elementary Probability for Applications__, Durrett. (or a more advanced text for someone who has at least taken 115a and 131a:) __Probability: Theory and Examples__, Durrett.**First Midterm:** Friday, April 28, 1PM-150PM, Broad 2100A**Second Midterm:** Monday May 22, 1PM-150PM, MS 5127**Final Exam:**Tuesday, June 13, 8AM-11AM, Boelter 2444**Other Resources:**Supplemental Problems from the textbook. An introduction to mathematical arguments, Michael Hutchings, An Introduction to Proofs, How to Write Mathematical Arguments**Email Policy:**

- My email address for this course is heilman(@-symbol)ucla.edu
- It is your responsibility to make sure you are receiving emails from heilman(@-symbol)ucla.edu , and they are not being sent to your spam folder.
- Do NOT email me with questions that can be answered from the syllabus.

**Exam Procedures:**Students must bring their UCLA ID cards to the midterms and to the final exam. Phones must be turned off. Cheating on an exam results in a score of zero on that exam. Exams can be regraded at most 15 days after the date of the exam. This policy extends to homeworks as well. All students are expected to be familiar with the UCLA Student Guide to Academic Integrity. If you are an OSD student, I would encourage you to discuss with me ways that I can improve your learning experience; I would also encourage you to contact the OSD office to confirm your exam arrangements at the beginning of the quarter.

**Exam Resources:**Here are the exams I used when I taught this course in previous quarters: Exam 1Exam 1 SolutionsExam 2Exam 2 Solutions. FinalFinal Solutions. Exam 1Exam 1 SolutionsExam 2Exam 2 Solutions. FinalFinal Solutions. Here is a page containing old exams for another 170A class. Here is a 170A practice midterm (with solutions). Here is a 170A practice second midterm. Here is a 170A practice final. Occasionally these exams will cover slightly different material than this class, or the material will be in a slightly different order, but generally, the concepts should be close if not identical.

**Homework Policy:**

- Late homework is not accepted.
- If you still want to turn in late homework, then the number of minutes late, divided by ten, will be deducted from the score. (The time estimate is not guaranteed to be accurate.)
- The lowest two homework scores will be dropped. This policy is meant to account for illnesses, emergencies, etc.
- Do not submit homework via email.
- There will be 8 homework assignments, assigned weekly on Tuesday and turned in at the
**beginning**of the discussion section on the following Tuesday. - A random subset of the homework problems will be graded each week. However, it is strongly recommended that you try to complete the entire homework assignment.
- You may use whatever resources you want to do the homework, including computers, textbooks, friends, the TA, etc. However, I would discourage any over-reliance on search technology such as Google, since its overuse could degrade your learning experience. By the end of the quarter, you should be able to do the entire homework on your own, without any external help..
- All homework assignments must be
**written by you**, i.e. you cannot copy someone else's solution verbatim. - Homework solutions will be posted on Friday after the homework is turned in.

**Grading Policy:**

- The final grade is weighted as the larger of the following two schemes. Scheme 1: homework (15%), the first midterm (20%), the second midterm (25%), and the final (40%). Scheme 2: homework (15%), largest midterm grade (35%), final (50%). The grade for the semester will be curved. However, anyone who exceeds my expectations in the class by showing A-level performance on the exams and homeworks will receive an A for the class.
- We will use the MyUCLA gradebook.
- If you cannot attend one of the exams, you must notify me within the first two weeks of the start of the quarter. Later requests for rescheduling will most likely be denied.
- You must attend the final exam to pass the course.

** Tentative Schedule**: (This schedule may change slightly during the course.)

Week | Monday | Tuesday | Wednesday | Thursday | Friday |

1 | Apr 3: 1.1, Sets | Apr 4: Homework 0 (ungraded) | Apr 5: 1.2, Probabilistic Models | Apr 7: 1.2, Probabilistic Models | |

2 | Apr 10: 1.3, Conditional Probability | Apr 11: Homework 1 due | Apr 12: 1.3, Conditional Probability | Apr 14: 1.4, Total Probability Theorem and Bayes' Rule | |

3 | Apr 17: 1.5, Independence | Apr 18: Homework 2 due | Apr 19: 1.5, Independence | Apr 21: 1.6, Counting | |

4 | Apr 24: 2.1, Discrete Random Variables | Apr 25: Homework 3 due | Apr 26: 2.2, Probability Mass Function | Apr 28: Midterm #1 | |

5 | May 1: 2.3, Functions of Random Variables | May 2: Homework 4 due | May 3: 2.4, Expectation and Variance | May 5, 2.5, Joint PMFs, Covariance and Variance | |

6 | May 8: 2.6, Conditioning | May 9, Homework 5 due | May 10: 2.6, Conditioning | May 12: 2.7, Independence | |

7 | May 15, 2.7, Independence | May 16: Homework 6 due | May 17: 3.1, Continuous random variables and PDFs | May 19: 3.1, Continuous random variables and PDFs | |

8 | May 22: Midterm #2 | May 23: No homework due | May 24: 3.2, Cumulative Distribution Functions | May 26: 3.3, Normal Random Variables | |

9 | No class | May 30: Homework 7 due | Jun 1: Joint PDFs of Multiple Random Variables | Jun 3: 3.5, Conditioning | |

10 | Jun 5: 3.5, Conditioning | Jun 6: Homework 8 due | Jun 7: The Continuous Bayes Rule | Jun 9, Review of Course |

**Advice on succeeding in a math class:**

- Review the relevant course material
**before**you come to lecture. Consider reviewing course material a week or two before the semester starts. - When reading mathematics, use a pencil and paper to sketch the calculations that are performed by the author.
- Come to class with questions, so you can get more out of the lecture. Also, finish your homework at least
**two days**before it is due, to alleviate deadline stress. - Write a rough draft and a separate final draft for your homework. This procedure will help you catch mistakes. Also, consider typesetting your homework. Here is a template .tex file if you want to get started typesetting.
- If you are having difficulty with the material or a particular homework problem, review Polya's Problem Solving Strategies, and come to office hours.

**Homework**

**Exam Solutions**

**Supplementary Notes**

Хейл высокомерно засмеялся. - Одна из проблем, связанных с приемом на работу самых лучших специалистов, коммандер, состоит в том, что иной раз они оказываются умнее. - Молодой человек, - вскипел Стратмор, - я не знаю, откуда вы черпаете свою информацию, но вы переступили все допустимые границы. Вы сейчас же отпустите мисс Флетчер, или я вызову службу безопасности и засажу вас в тюрьму до конца ваших дней.

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