master-hwsol.pdf

# master-hwsol.pdf - 36-217 Probability Theory and Random...

• Notes
• 84

This preview shows page 1 - 4 out of 84 pages.

36-217 Probability Theory and Random Processes Homework Solutions Sangwon (Justin) Hyun June 22, 2016 Homework 1 (due Thursday May 19th) 1. (25 pts) Consider the Normal distribution N ( μ, σ 2 ) that we mentioned in class and a sample X 1 , . . . , X n of size n obtained from this distri- bution. Which of the following are statistics and why? (a) ¯ X (b) n ( ¯ X - μ ) S (c) Q n i =1 X i (d) max { X 1 , . . . , X n , σ 2 } (e) { X 3 > 0 } ( X 3 ) Extra credit (5pt) What parameter do you think ¯ X is trying to estimate? Justify your answer. Solution: (a), (c), and (e) are statistics since they are functions de- pending only on the data. (b) and (d) are not statistics since they also depend on the parameters μ and σ 2 . Extra Credit: The sam- ple mean is a good estimate of the actual mean of the distribution μ . This is because, intuitively, we can expect the sample mean of many repeated experiments to well represent the underlying process. We will learn more about this when we learn about the central limit theorem . 2. (25 pts) Show that, for any A, B 1 , . . . , B n Ω with n i =1 B i = Ω, we have A = ( A B 1 ) ∪ · · · ∪ ( A B n ). Solution: We have A = A Ω and Ω = n i =1 B n . Thus, A = A Ω = A ( n i =1 B i ) 1
Using the distributive property of intersection, we get A = n i =1 ( A B i ) . 3. (25 pts) Consider the following experiments: you have three keys and exactly one of them opens the door that you want to open; at each attempt to open the door, you pick one of the keys completely at random and try to open the door (at each attempt you pick one of the three keys at random, regardless of the fact that you may already have tried some in previous attempts). You count the number of attempts needed to open the door. What is the sample space for this experiment? you have an urn with 1 red ball, 1 blue ball and 1 white ball. For two times, you draw a ball from the urn, look at its color, take a note of the color on a piece of paper, and put the ball back in the urn. What is the sample space for this experiment? Write out the set corresponding to the event ‘you observe at least 1 white ball’. Solution: The sample space is the set of positive integers Ω = { 1 , 2 , 3 , 4 , 5 , . . . } . Let R: red, B: blue, and W: white. The sample space is the set of all possible pairs of observed colors Ω = { RR, RB, RW, BR, BB, BW, WR, WB, WW } . Let A be the event ‘you observe at least 1 white ball’. Then A = { RW, BW, WR, WB, WW } . 4. (25 pts) Think of an example of a random process and describe what the quantity of interest X t is and what the variable t represents in your example (time, space, . . . ?) Solution: Graded according to how specific/believable the example is. 5. Extra credit (10 pts) Show that the expression of the sample variance S 2 = 1 n - 1 n X i =1 ( X i - ¯ X ) 2 (1) 2
is equivalent to S 2 = 1 n - 1 n X i =1 X 2 i - 1 n n X i =1 X i ! 2 . (2) Solution: We have ( n - 1) S 2 = n X i =1 ( X i - ¯ X ) 2 = n X i =1 ( X 2 i - 2 X i ¯ X + ¯ X 2 ) = n X i =1 X 2 i - 2 ¯ X n X i =1 X i + n ¯ X 2 = n X i =1 X 2 i - 2 n ¯ X 2 + n ¯ X 2 = n X i =1 X 2 i - n ¯ X 2 = n X i =1 X 2 i - n 1 n 2 n X i =1 X i !

Want to read all 84 pages?

Want to read all 84 pages?

#### You've reached the end of your free preview.

Want to read all 84 pages?

{[ snackBarMessage ]}

###### "Before using Course Hero my grade was at 78%. By the end of the semester my grade was at 90%. I could not have done it without all the class material I found."
— Christopher R., University of Rhode Island '15, Course Hero Intern

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern