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f2011homework14-2 - (c C = cfw_4 Chapter 1 Homework FALL...

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Chapter 1 Homework; FALL 2011 1. Consider the following CM: A 10-sided die is tossed, with faces marked 1, 2, . . . , 10. The outcome is the number on the face that lands up. (a) Determine the sample space. (b) List the elements of the following event: A = the outcome is an odd number. (c) List the elements of the following event: B = the outcome is an even number larger than 6. (d) Describe the following event in words: C = { 7 , 8 , 9 , 10 } . 2. Refer to the previous exercise. Assume the ELC. Calculate the probability of each of the events, A , B and C . 3. A CM has a sample space that consists of five outcomes: 1, 2, 3, 4 and 5. For each of the following assignments, decide whether it is a mathematically valid way to assign probabilities for this situation. If not, ex- plain why not. (a) P (1) = 0 . 30 , P (2) = 0 . 15 , P (3) = 0 . 25 , P (4) = 0 . 20 , P (5) = 0 . 10 . (b) P (1) = 0 . 30 , P (2) = 0 . 15 , P (3) = 0 . 20 , P (4) = 0 . 20 , P (5) = 0 . 10 . (c) P (1) = 0 . 30 , P (2) = 0 . 15 , P (3) = 0 . 25 , P (4) = 0 . 20 , P (5) = 0 . 20 . (d) P (1) = 0 . 30 , P (2) = 0 . 15 , P (3) = 0 . 25 , P (4) = 0 . 40 , P (5) = - 0 . 10 . 4. Refer to the previous exercise. Use assign- ment (a) to calculate the probability of each of the following events. 5. Refer to the previous exercise. (a) Verify that: P ( B or D ) = P ( B ) + P ( D ) . (b) Verify that Rule 6 is true for events B and D . (c) Given that P ( B ) = 0 . 15 + 0 . 20 = 0 . 35 , explain why you know that P ( A ) = 0 . 65 without adding the probabilities of outcomes 1, 3 and 5. (d) Of the five events listed in Exercise 4, find all pairs that illustrate Rule 5. 6. You are given the following information: the events A and B are disjoint; P ( A ) = 0 . 30 ; and P ( B ) = 0 . 55 . Calculate the fol- lowing probabilities. (a) P ( A or B ) . (b) P ( A c ) . (c) P ( B c ) . 7. You are given the following information: P ( A ) = 0 . 65 ; P ( B ) = 0 . 45 ; P ( AB ) = 0 . 30 . Calculate P ( A or B ) . 1
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Chapter 1 Homework Continued 8. Consider a sample space with five mem- bers: 0, 1, 2, 3 and 4. Assume the ELC and i.i.d. trials. Define X = X 1 + X 2 , the to- tal of the numbers obtained in the first two trials. Find the sampling distribution of X . 9. Consider a sample space with three mem- bers: 1, 2 and 3. Assume the ELC and i.i.d. trials. Define X = X 1 + X 2 + X 3 , the total of the numbers obtained in the first three tri- als. Find the sampling distribution of X . 10. Consider a sample space with four mem- bers: 1, 2, 3 and 4. Do not assume the ELC. Instead assume the following: P (1) = 0 . 1 , P (2) = 0 . 2 , P (3) = 0 . 3 and P (4) = 0 . 4 . Assume i.i.d. trials. Define X = X 1 + X 2 , the total of the numbers obtained in the first two trials. Find the sampling distribution of X . 11. Refer to the previous question. Define X = X 1 + X 2 + X 3 , the total of the numbers obtained in the first three trials. Find the sampling distribution of X . 12. Refer to the table in problem 10 of the Chapter 1 Lecture Examples. (a) Verify (i.e. count 5-tuples and then di- vide) my statement that P ( X = 7) = 0 . 00193 . (b) Verify (i.e. count 5-tuples and then divide) my statement that P ( X = 27) = 0 . 00450 . (c) Verify the nearly certain interval I give for P ( X = 17) .
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