The two parameter gamma distribution can be generalized by introducing a third parameter , called a threshold or location parameter replace x in (4 8) by x and x 0 by x This amounts to shifting the density curves in Figure 4 27 so that they begin their ascent or descent at rather than 0 The article Bivariate Flood Frequency Analysis with Historical Information Based on Copulas (J of Hydrologic Engr , 2013 1018 1030) employs this distribution to model X 3 day flood volume (108 m3) Suppose that values of the parameters are 12, 7, 40 (very close to estimates in the cited article based on past data) a What are the mean value and standard deviation of X b What is the probability that flood volume is between 100 and 150 c What is the probability that flood volume exceeds its mean value by more than one standard deviation d What is the 95th percentile of the flood volume distribution

Question: The two-parameter gamma distribution can be generalized by introducing a third parameter , called a threshold or location parameter: replace x in (4.8) by x

The two-parameter gamma can be generalized by introducing a third parameter γ, called a threshold or location parameter: replace x in (4.8) by x - γ and x ≥ 0 by x ≥ γ. This amounts to shifting the density curves in Figure 4.27 so that they begin their ascent or descent at γ rather than 0. The article "Bivariate Flood Frequency Analysis with Historical Information Based on Copulas" (J. of Hydrologic Engr., 2013: 1018-1030) employs this to model X = 3-day flood volume (108 m3). Suppose that values of the parameters are α = 12, β = 7, γ = 40 (very close to estimates in the cited article based on past data).
a. What are the mean value and standard deviation of X?
b. What is the probability that flood volume is between 100 and 150?
c. What is the probability that flood volume exceeds its mean value by more than one standard deviation?
d. What is the 95th percentile of the flood volume

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