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The two-parameter gamma distribution can be generalized by introducing a third parameter y, called a threshold or location parameter: replace x in (4.8), x 20 f(x; a) = r(a) (4.7) otherwise by x - y and x 2 0 by x 2 y. This amounts to shifting the density curves in the figure below so that they begin their ascent or descent at y rather than 0. f(x; a, B) 4 f(x; a.) +

The two-parameter gamma distribution can be generalized by introducing a third parameter y, called a threshold or location parameter: replace x in (4.8), x 20 f(x; a) = r(a) (4.7) otherwise by x - y and x 2 0 by x 2 y. This amounts to shifting the density curves in the figure below so that they begin their ascent or descent at y rather than 0. f(x; a, B) 4 f(x; a.) +

A First Course in Probability (10th Edition)
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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The two-parameter gamma distribution can be generalized by introducing a third parameter y, called a threshold or location parameter: replace x in (4.8), (xa-le-x x2 0 f(x; a) = r(a) (4.7) otherwise by x - y and x 2 0 by x 2 y. This amounts to shifting the density curves in the figure below so that they begin their ascent or descent at y rather than 0. f(x; a, ß) 4 f(x; a,) - a = 2, B = 1.0 - 1.0 – a = 1 a = 1, B = 1 0.5 - a = .6 a = 2, ß = 2 0.5 - a = 2 a = 5 a = 2, B = 1 1 2 3 4 6. 7 3 4. 5 (a) gamma density curves (b) standard gamma density curves A study employs this distribution to model X = 3-day flood volume (10° m3). Suppose that values of the parameters are a = 12, B = 8, y = 38 (very close to estimates in the cited article based on past data). (a) What are the mean value and standard deviation of X? (Round your answers to four decimal places.) | 10°m3 | 10°m3 mean standard deviation (b) What is the probability that flood volume is between 100 and 156? (Round your answer to three decimal places.) (c) What is the probability that flood volume exceeds its mean value by more than one standard deviation? (Round your answer to three decimal places.) (d) What is the 95th percentile of the flood volume distribution? (Round your answer to two decimal places.) 10°m3
Transcribed Image Text:The two-parameter gamma distribution can be generalized by introducing a third parameter y, called a threshold or location parameter: replace x in (4.8), (xa-le-x x2 0 f(x; a) = r(a) (4.7) otherwise by x - y and x 2 0 by x 2 y. This amounts to shifting the density curves in the figure below so that they begin their ascent or descent at y rather than 0. f(x; a, ß) 4 f(x; a,) - a = 2, B = 1.0 - 1.0 – a = 1 a = 1, B = 1 0.5 - a = .6 a = 2, ß = 2 0.5 - a = 2 a = 5 a = 2, B = 1 1 2 3 4 6. 7 3 4. 5 (a) gamma density curves (b) standard gamma density curves A study employs this distribution to model X = 3-day flood volume (10° m3). Suppose that values of the parameters are a = 12, B = 8, y = 38 (very close to estimates in the cited article based on past data). (a) What are the mean value and standard deviation of X? (Round your answers to four decimal places.) | 10°m3 | 10°m3 mean standard deviation (b) What is the probability that flood volume is between 100 and 156? (Round your answer to three decimal places.) (c) What is the probability that flood volume exceeds its mean value by more than one standard deviation? (Round your answer to three decimal places.) (d) What is the 95th percentile of the flood volume distribution? (Round your answer to two decimal places.) 10°m3
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