A fair die is rolled whose six sides are labeled 1, 2, 2, 3, 3, 3. Let D be the number that comes up. (a) Find the z-transform Db (z), defined to be E[z^D]. (b) Let D1, D2, D3, D4 be four independent copies of D (that is, the results of four rolls of the die), and let S = D1 + D2 + D3 + D4. Find the z-transform Sb(z). (Don’t expand.) (c) Wolfram Alpha told me that when you expand the answer to (b), you should get z^4/1296 + z^5/162 + z^6/36 + 13z^7/162 + 107z^8/648 + 13z^9/54 + z^10/4 + z^11/6 + z^12/16. Using this information, what is Pr[S ≥ 10]?

A fair die is rolled whose six sides are labeled 1, 2, 2, 3, 3, 3. Let D be the number that comes up. (a) Find the z-transform Db (z), defined to be E[z^D]. (b) Let D1, D2, D3, D4 be four independent copies of D (that is, the results of four rolls of the die), and let S = D1 + D2 + D3 + D4. Find the z-transform Sb(z). (Don’t expand.) (c) Wolfram Alpha told me that when you expand the answer to (b), you should get z^4/1296 + z^5/162 + z^6/36 + 13z^7/162 + 107z^8/648 + 13z^9/54 + z^10/4 + z^11/6 + z^12/16. Using this information, what is Pr[S ≥ 10]?

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

A fair die is rolled whose six sides are labeled 1, 2, 2, 3, 3, 3. Let D be the number that comes
up.
(a) Find the z-transform Db (z), defined to be E[z^D].
(b) Let D1, D2, D3, D4 be four independent copies of D (that is, the results of four rolls of
the die), and let S = D1 + D2 + D3 + D4. Find the z-transform Sb(z). (Don’t expand.)
(c) Wolfram Alpha told me that when you expand the answer to (b), you should get
z^4/1296 + z^5/162 + z^6/36 + 13z^7/162 + 107z^8/648 + 13z^9/54 + z^10/4 + z^11/6 + z^12/16.
Using this information, what is Pr[S ≥ 10]?

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Step 2: Compute the z-transform Db(z), defined to be E[z^D].

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Step 3: Compute the z-transform Sb(z) four independent copies of D.

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Step 4: Compute the value of Pr[S ≥ 10] using the given information.

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