The two-parameter gamma distribution can be generalized by introducing a third parameter \( \gamma \), called a threshold or location parameter: replace \( x \) in (4.8),\[f(x; \alpha) = \begin{cases} \frac{x^{\alpha -1} e^{-x}}{\Gamma(\alpha)} & x \geq 0 \\0 & \text{otherwise} \end{cases}\](4.7)by \( x - \gamma \) and \( x \geq \gamma \). This amounts to shifting the density curves in the figure below so that they begin their ascent or descent at \( y \) rather than 0.### Graphs:(a) **Gamma Density Curves**- Displays curves for \( \alpha = 2, \beta = \frac{1}{3} \); \( \alpha = 1, \beta = 1 \); \( \alpha = 2, \beta = 2 \); \( \alpha = 2, \beta = 1 \).- The x-axis is labeled \( x \).- The y-axis is labeled \( f(x; \alpha, \beta) \).(b) **Standard Gamma Density Curves**- Displays curves for \( \alpha = 1 \); \( \alpha = 0.6 \); \( \alpha = 2 \); \( \alpha = 5 \).- The x-axis is labeled \( x \).- The y-axis is labeled \( f(x; \alpha) \).### Study Details:A study employs this distribution to model \( X = \) 3-day flood volume \((10^8 \, m^3)\). Suppose that values of the parameters are \( \alpha = 11 \), \( \beta = 6 \), \( \gamma = 43 \) (very close to estimates in the cited article based on past data).#### Questions:(a) **What are the mean value and standard deviation of \( X \)? (Round your answers to four decimal places.)**- **Mean:** \[ 115 \times 10^8 m^3 \]- **Standard deviation:** \[ 20.7846 \times 10^8 m^3 \](b) **What is the probability that flood volume is between 100 and 152? (Round your answer to three decimal places.)

Given: α=11, β=6, γ=43 (a) The objective is to determine the mean value and standard deviation of X.…

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-1-x I(a) 0 xz0 otherwise by x - y and x > 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) f(x, α₂) 4 a=1 Kar a=6 0.5- a=2 2 3 (b) standard gamma density curves f(x: a) = mean 1.0- 0.5- a=2₁B = } a=1, B=1 2 (a) gamma density curves standard deviation T 20.7846 (4.7) a=2, B=2 4 _a=2, B=1 T 6 7 x 1.0- A study employs this distribution to model X = 3-day flood volume (108 m³). Suppose that values of the parameters are a = 11, ß = 6, y = 43 (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.) 115 X 108m³ x 108m³ 0 a = 5 (b) What is the probability that flood volume is between 100 and 152? (Round your answer to three decimal places.) 0.096 x (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.) 0.005 x (d) What is the 95th percentile of the flood volume distribution? (Round your answer to two decimal places.)

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...