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

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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
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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
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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 !
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