Homework 1. Spring 2019 (Due Tuesday January 22)

Transcription

1 ECE 302: Probabilistic Methods in Electrical and Computer Engineering Spring 2019 Instructor: Prof. A. R. Reibman Homework 1 Spring 2019 (Due Tuesday January 22) Homework is due on Tuesday January 22 at the beginning of class. No late homework will be accepted. Include a brief description of all sources of information you used (including other people), not counting the text, handouts, or material posted on the web page, or state I did not receive help on this homework. You do not need to reference any material presented in class or on the course web-site, in the textbook, nor Prof. Reibman nor TA Chen Bai. Topics: Sample space, events, set theory, Probability mappings; reading in Chapters 1 and Section Conditional Probability (Section 2.4) Exercise 1. The space S and three of its subsets are given by S = {n Z : 0 n 11}, A = {1, 3, 4, 5, 9}, and B = {4, 7, 9, 11}, and C = {1, 3, 9, 11}. Find A B C, A c B, A C, and (A B) B.

2 Exercise 2. (from exam 1, Fall 2015) For each of the following relations, determine which is valid for arbitrary events A, B, and C. (Note: to be true for arbitrary events, it must be true for any such event. Use a Venn diagram if it is helpful.) (On the exam, this was a True/False question. For this homework, show whether it is true or false.) (a) (A B C) c = A c B c C c (b) (A B) (A B c ) (A c B) = (A c B c ) c (c) (A B) C = A (B C). (d) (A B) (A c B c ) = (A B c ) (A c B) (A c B C c ) Hint: be careful, item (d) is tricky. 2

3 Exercise 3. Four marbles, numbered 1,2,3 and 4 are placed in a box. One of the marbles is drawn randomly from the box and its number, N 1 is noted. (So, N 1 =1,2,3, or 4.) An integer N 2 is then selected at random from the values 1,..., N 1. The outcome of this experiment is the ordered pair (N 1, N 2 ), where N 1 denotes the marble and N 2 is just a number. (a) Write the sample space of the experiment. (b) Write the event Marble 2 is selected. (c) Write the event N 2 = 3. (d) Write the event Marble 2 is selected and N 2 = 3. 3

4 Exercise 4. An integrated circuit (IC) factory has three machines, X, Y, and Z. Test one IC from each machine, and observe if each is acceptable (a) or fails (f). Thus, an observation is a sequence of the three test results from each machine. For example, the observation that the circuit from Z fails while the circuits from X and Y pass is aaf. (a) What is the sample space? (b) What are the elements of the sets Z F = {circuit from Z fails} and X A = {circuit from X is acceptable}? (c) Are Z F and X A mutually exclusive? Are Z F and X A collectively exhaustive? (d) What are the elements of the sets C = {more than one circuit is acceptable} and D = {at least two circuits fail}? (e) Are C and D mutually exclusive? Are C and D collectively exhaustive? 4

5 Exercise 5. Similar to but not textbook problem 2.38 Two numbers (x, y) are selected at random from the interval [0, 1]. (a) Find the probability that the pair of numbers are outside the unit circle. (b) Find the probability that 3y > x. 5

6 Exercise 6. Mobile phones perform handoffs as they move from one cell to another. During a call, a phone either performs zero handoffs (H 0 ), one handoff (H 1 ), or more than one handoff (H 2 ). In addition, each call is either long (L), if it last more than three minutes, or short S. The following table describes the probabilities of the possible types of calls. H 0 H 1 H 2 L S Complete the table by finding the missing value. What is the probability P (H 0 ) that a phone makes no handoffs? What is the probability a call is short? What is the probability a call is long or there are at least two handoffs? 6

7 Exercise 7. (From Exam 1, Fall 2016) Among the Purdue students taking ECE 302 this semester, some like dogs, some like cats, some like both, and some like neither. Let D be the set of Purdue ECE 302 students who like dogs, and C be the set who like cats. A study shows that 22% like both cats and dogs, and 12% like neither. The probability a student likes dogs exceeds the probability a student likes cats by What is the probability a randomly chosen student likes cats? 7

8 Exercise 8. Suppose among all six-letter English words, a word is picked at random (i.e., each six-letter word has the same probability of being picked). Which event is more probable: That the selected word has an n as its fifth letter, or that the selected word ends in ing? 8

9 Exercise 9. From Exam 1, Fall 2015 You have 4 otherwise identical cans of soda (also known as pop or cola), except you know that 1 was shaken up about 10 minutes ago, while the other 3 have been stable for hours. (You have lost track of which can is which.) The probability of the shaken can splattering when opened is 4/5, and the probability of a stable can splattering when opened is 1/3. (a) If you choose one can at random and open it, what is the probability of it splattering? (b) If you open a can and it splatters, what is the probability that it was the shaken can? (You may leave your answers in fractional form.) 9

10 Exercise 10. A crime is committed on an island with a population of A priori, each person is equally likely to have committed the crime. Based on a forensic test, a suspect is arrested. This is the only evidence that is presented at trial. The accuracy of the test is 99.9%, that is, P(test is positive suspect is guilty) = 0.999, and P(test is negative suspect is innocent) = You are on the jury. Do you have reasonable doubt that this person committed the crime? (Hint, find the probability the person is guilty given that they received a positive test.) 10