13. For each of the collections of the subsets of 2 = {W1, W2, W3, W4} set out below, statewhether it is a σ-algebra. If it is not, explain why.(a) F₁ = P(N).(b) F₂ = {0, N}.(c) F3 = {0, {w1}, {W2, W3}, {w₁}, }.(d) F₁ = {0, {w1}, {W2, W3, W1}, N}.14. Let N = {w1, W2, W3, W1}. A random variance X : → R hasX=1 in state wi3 in state W3-2 in states W2, W4Let F={0, {w1, w₂}, {w3, W₁}, }.(a) Is F is a σ-algebra? Give a brief explanation for your answer.(b) Is X F-measurable? Give a brief explanation for your answer.

Step 1: 13. To check if each collection is a σ-algebra, we need to check three properties:It…

Answered: 13. For each of the collections of the…

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter12: Queueing Models

Section12.3: The Exponential Distribution

Problem 3P

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In this version of dice blackjack, you toss a single die repeatedly and add up the sum of your dice tosses. Your goal is to come as close as possible to a total of 7 without going over. You may stop at any time. If your total is 8 or more, you lose. If your total is 7 or less, the house then tosses the die repeatedly. The house stops as soon as its total is 4 or more. If the house totals 8 or more, you win. Otherwise, the higher total wins. If there is a tie, the house wins. Consider the following strategies: Keep tossing until your total is 3 or more. Keep tossing until your total is 4 or more. Keep tossing until your total is 5 or more. Keep tossing until your total is 6 or more. Keep tossing until your total is 7 or more. For example, suppose you keep tossing until your total is 4 or more. Here are some examples of how the game might go: You toss a 2 and then a 3 and stop for total of 5. The house tosses a 3 and then a 2. You lose because a tie goes to the house. You toss a 3 and then a 6. You lose. You toss a 6 and stop. The house tosses a 3 and then a 2. You win. You toss a 3 and then a 4 for total of 7. The house tosses a 3 and then a 5. You win. Note that only 4 tosses need to be generated for the house, but more tosses might need to be generated for you, depending on your strategy. Develop a simulation and run it for at least 1000 iterations for each of the strategies listed previously. For each strategy, what are the two values so that you are 95% sure that your probability of winning is between these two values? Which of the five strategies appears to be best?

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