Publications by Marley Myrianthopoulos
DATA 605 Discussion 10
Page 423 Problem 9 Task: A process moves on the integers 1, 2, 3, 4, and 5. It starts at 1 and, on each successive step, moves to an integer greater than its present position, moving with equal probability to each of the remaining larger integers. State five is an absorbing state. Find the expected number of steps to reach state five. Solution: T...
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DATA 605 HW 10
This assignment uses random number generation, so I am setting a seed for consistent results: set.seed(149162536) Task: Smith is in jail and has 1 dollar; he can get out on bail if he has 8 dollars. A guard agrees to make a series of bets with him. If Smith bets A dollars, he wins A dollars with probability .4 and loses A dollars with probability ...
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DATA 605 HW 9
Problem 1 Task: The price of one share of stock in the Pilsdorff Beer Company is given by \(Y_n\) on the \(n\)th day of the year. Finn observes that the differences \(X_n=Y_{n+1}-Y_n\) appear to be independent random variables with a common distribution having mean \(\mu=0\) and variance \(\sigma^2=1/4\). If \(Y_1=100\), estimate the probability t...
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DATA 605 Discussion 9
Page 338 Exercise 3 Task: A true-false examination has 48 questions. June has probability 3/4 of answering a question correctly. April just guesses on each question. A passing score is 30 or more correct answers. Compare the probability that June passes the exam with the probability that April passes it. We can model each of these as a binomial di...
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Data 605 HW 8
Page 303 Problem 11 Task: 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? Solution: If each lightbulb’s lifespan is modeled as an independent random variable \(X_1, X_2, ...,X_{100}\) then each of them has an exponential density with mean...
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Data 605 Discussion 8
Page 312 #1 Task: A fair coin is tossed 100 times. The expected number of heads is 50, and the standard deviation for the number of heads is \((100 \cdot 1/2 \cdot 1/2)^{1/2}=5\). What does Chebychev’s inequality tell you about the probability that the number of heads that turn up deviates from the expected number of 50 by three or more standard...
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Data 605 HW 7
Problem 1: Task: 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 . Solution: The probability that \(Y=y\) is the probability that none of the variables is less than \(y\) AND that the variables ...
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Data 605 Discussion 7
Page 199 #14 On the average, only 1 person in 1000 has a particular rare blood type. (a) Find the probability that, in a city of 10,000 people, no one has this blood type. The probability can be calculated as \(P=0.999^{10000}=0.0000451733\) So the probability that no one in the city has this blood type is approximately 0.0045%. (b) How many peop...
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Data 605 HW 6
Problem 1: Task: 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? Solution: There are \({7 \choose 5}=\frac{7!}{2!5!}=21\) ways of drawing only red jellybeans, and \({5 \choose 1}{7 \choose 4}=\frac{5!}{4!1!}\frac{7!}{3!4!}=(5)(35)...
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Data 605 Discussion 5
Page 198 #9 A worker for the Department of Fish and Game is assigned the job of estimating the number of trout in a certain lake of modest size. She proceeds as follows: She catches 100 trout, tags each of them, and puts them back in the lake. One month later, she catches 100 more trout, and notes that 10 of them have tags. (a) Without doing any f...
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