Publications by Kristin Lussi
DATA 605 Homework 9
Homework 9 Exercise 1 The price of one share of stock in the Pilsdorff Beer Company (see Exercise 8.2.12) 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 ...
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DATA 605 Week 9 Discussion
Page 354 #5 A die is thrown until the first time the total sum of the face values of the die is 700 or greater. Estimate the probability that, for this to happen, more than 210 tosses are required less than 190 tosses are required between 180 and 210 tosses, inclusive, are required Solution First, we will find the expected value and variance. \[ ...
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DATA 605 Homework 8
Page 303 Exercise 11 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.) Solution by Hand For any of the bulbs, $X_i = $ its independent random variable. Let \(\Sigma X_i = min(x_1, x_2, ... x_{100})\). Let \(n=100\). From the...
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DATA 605 Week 8 Discussion
Page 302 #5 Suppose that X and Y are independent and \(Z = X + Y\). Find \(f_Z\) if Part A: \[ f_x(x) = \begin{cases} \lambda^{-\lambda x} \text{ if } x>0\\ 0 \text{ otherwise}\\ \end{cases}\\ f_y(x) = \begin{cases} \mu e^{-\mu y} \text{ if } x>0\\ 0 \text{ otherwise} \end{cases} \] Solution: \[ f_z(Z) = \int_{-\infty}^{\infty} f_x (z-y) f_y(y)dy...
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DATA 605 Homework 7
Homework 7 Exercise 1 Let \(X_1, X_2, . . . , X_n\) 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 \(X_i’\)s. Find the distribution of \(Y\) . Solution \[ \text{We are trying to find that the minimum of } X_1, X_2, . . . , X_n \text{is l...
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DATA 605 Discussion Week 7
Page 248 #8 A royal family has children until it has a boy or until it has three children, whichever comes first. Assume that each child is a boy with probability 1/2. Find the expected number of boys in this royal family and the expected num- ber of girls. Solution by Hand We’ll let \(X = \text{Boys}\) and \(Y = \text{Girls}\). \[ P(X=1, Y=0) =...
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DATA 605 Week 6 Homework
Exercise 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? Solution: P(Green <2) = P(4 red & 1 green) + P(5 red & 0 green) \[ \text{green }<2 = 4 \text{ red & } 1 \text{ green} + 5 \text{ red & } 0 \text{ green}\\ 4 \text{ red & } ...
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DATA 605 Week 6 Discussion
Page 115 #19 A gin hand consists of 10 cards from a deck of 52 cards. Find the probability that a gin hand has a) all 10 cards of the same suit. b) exactly 4 cards in one suit and 3 in two other suits. c) a 4, 3, 2, 1, distribution of suits. a) all 10 cards of the same suit \[ \text{Ways to choose 10 cards: } {52\choose 10}\\ \text{Ways to choose ...
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DATA 605 Week 5 Homework
Week 5 Homework Exercise 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 th...
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DATA 605 Week 5 Discussion
Page 199 Exercise 13 The poisson distribution with parameter \(\lambda = .3\) has been assigned for the outcome of an experiment. Let \(X\) be the outcome function. Find \(P(X=0)\), \(P(X=1)\), and \(P(X>1)\) Solution by Hand Poisson Formula: \[ P(X=x) = \frac{\lambda^x e^{-\lambda}}{x!} \] \(P(X=0)\) \[ P(X=0) = \frac{0.3^0 e^{-0.3}}{0!} = \fra...
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