Publications by Ariann Chai
Data 605 Discussion Post 9
Q11 p355 A tourist in Las Vegas was attracted by a certain gambling game in which the customer stakes 1 dollar on each play; a win then pays the customer 2 dollars plus the return of her stake, although a loss costs her only her stake. Las Vegas insiders, and alert students of probability theory, know that the probability of winning at this game is...
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Data 605 Assignment 9
Question 1 The price of one share of stock in the Pilsdorff Beer Company (see Exercise 8.2.12) is given by Yn on the nth day of the year. Finn observes that the differences Xn = Yn+1 − Yn appear to be independent random variables with a common distribution having mean µ = 0 and variance \(σ^2 = 1/4\). If Y1 = 100, estimate the probability that ...
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Data 605 Discussion Post 8
Q5 p290 Consider the following two experiments: the first has outcome X taking on the values 0, 1, and 2 with equal probabilities; the second results in an (independent) outcome Y taking on the value 3 with probability 1/4 and 4 with probability 3/4. Find the distribution of Y + X Y + X could be the values {3,4,5,6} For P(Y + X = 3): P(X = 0)P(Y =...
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Data 605 Assignment 8
Question 11 p303 A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out? (See Exercise 10.) Exercise 10 tells us that “Let M be the minimum value of the Xj . Show that the density for M is exponential with mean \(µ/n\)” so for us to find the exp...
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Data 605 Assignment 7
Question 1 Let X1, X2, . . . , Xn be n mutually independent random variables, each of which is uniformly distributed on the integers from 1 to k. Let Y denote the minimum of the Xi’s. Find the distribution of Y. Since the integers range from 1 to k, each of the n mutually independent random variables has k possible values. So, for the whole set o...
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Data 605 Discussion Post 7
Q4&5 p279 Question 4 In Las Vegas the roulette wheel has a 0 and a 00 and then the numbers 1 to 36 marked on equal slots; the wheel is spun and a ball stops randomly in one slot. When a player bets 1 dollar on a number, he receives 36 dollars if the ball stops on this number, for a net gain of 35 dollars; otherwise, he loses his dollar bet. Find t...
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Data 605 Discussion Post 6
Q15 p90 A computing center has 3 processors that receive n jobs, with the jobs assigned to the processors purely at random so that all of the \(3^n\) possible assignments are equally likely. Find the probability that exactly one processor has no jobs. To choose the processor with no job we do: \({3}\choose{1}\) With two processors left to do the wo...
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Data 605 Assignment 6
Question 1 A bag contains 5 green and 7 red jellybeans. How many ways can 5 jellybeans be withdrawn from the bag so that the number of green ones withdrawn will be less than 2? We need to find the number of ways for two possible outcomes that will fit this statement (# of green jellybeans < 2): If the 5 jellybeans are 0 green jellybean & 5 red jell...
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Data 605 Assignment 5
Question 1 (Bayesian). A new test for multinucleoside-resistant (MNR) human immunodeficiency virus type 1 (HIV-1) variants was recently developed. The test maintains 96% sensitivity, meaning that, for those with the disease, it will correctly report “positive” for 96% of them. The test is also 98% specific, meaning that, for those without the d...
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Data 605 Discussion Post 5
Q8 p198 If a coin is tossed a sequence of times, what is the probability that the first head will occur after the fifth toss, given that it has not occurred in the first two tosses? (Assuming that this is a fair coin as it would be stated otherwise.) Since its given that it has not occurred in the first two tosses, we can assume those two are tails...
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