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-¹e-x Γ(α) 0 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) 1.0- f(x, a.) 4 1.0- f(x; a) = 0.5- 0 x 20 otherwise a=2,B=₁ 1 2 (a) gamma density curves a=1, B=1 a=2, B=2 3 (4.7) 4 a=2, B = 1 S 6 7 X 0.5- 0 1 a=1 a=6 3 a=2 4 5 (b) standard gamma density curves (b) What is the probability that flood volume is between 100 and 154? (Round your answer to three decimal places.) 0.708 x a=5 A study employs this distribution to model X = 3-day flood volume (108 m³). Suppose that values of the parameters are a = 11, ß = 8, y = 41 (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.) mean 129 ✓ 108 m³ 108 m3 standard deviation 26.5330 (d) What is the 95th percentile of the flood volume distribution? (Round your answer to two decimal places.) 176.70 ✓ 108 m³ 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.) 0.156
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-¹e-x Γ(α) 0 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) 1.0- f(x, a.) 4 1.0- f(x; a) = 0.5- 0 x 20 otherwise a=2,B=₁ 1 2 (a) gamma density curves a=1, B=1 a=2, B=2 3 (4.7) 4 a=2, B = 1 S 6 7 X 0.5- 0 1 a=1 a=6 3 a=2 4 5 (b) standard gamma density curves (b) What is the probability that flood volume is between 100 and 154? (Round your answer to three decimal places.) 0.708 x a=5 A study employs this distribution to model X = 3-day flood volume (108 m³). Suppose that values of the parameters are a = 11, ß = 8, y = 41 (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.) mean 129 ✓ 108 m³ 108 m3 standard deviation 26.5330 (d) What is the 95th percentile of the flood volume distribution? (Round your answer to two decimal places.) 176.70 ✓ 108 m³ 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.) 0.156
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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-¹e-x
Γ(α)
0
f(x; a) =
mean
by x - y and x ≥ 0 by x ≥ 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)
1.0-
0.5-
0
ΧΣ Ο
otherwise
-α = 2, B =
a = 1, B=1
T
12
(a) gamma density curves
(4.7)
,α = 2, B = 2
a=2, B=1
T
T
T T T
3 4 5 6 7
X
1.0
a = 1
K
₂α = .6
0.5-
a=2
2 3 4 5
(b) standard gamma density curves
f(x; a,)
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 = 8, y = 41 (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.)
129
✓ 108 m³
3
standard deviation 26.5330
✓ 108 m³
(b) What is the probability that flood volume is between 100 and 154? (Round your answer to three decimal places.)
0.708
X
α = 5
X
(d) What is the 95th percentile of the flood volume distribution? (Round your answer to two decimal places.)
176.70
✓ 108 m³
(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.)
0.156
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It says part B is still wrong, I tried 0.2809 and 0.281, it says both of them are wrong. Please help.
![**Gamma Distribution with Threshold Parameter**
The two-parameter gamma distribution can be generalized by introducing a third parameter \(\gamma\), called a threshold or location parameter. This modifies the function as shown below:
\[
f(x; \alpha) =
\begin{cases}
\frac{x^{\alpha-1}e^{-x}}{\Gamma(\alpha)} & x \ge 0 \\
0 & \text{otherwise}
\end{cases}
\]
This adjustment amounts to shifting the density curves so they begin their ascent or descent at \(\gamma\) rather than 0.
**(a) Gamma Density Curves Diagram**
This graph illustrates several gamma density curves with different parameters \(\alpha\) and \(\beta\):
- \(\alpha = 2, \beta = \frac{1}{3}\)
- \(\alpha = 1, \beta = 1\)
- \(\alpha = 2, \beta = 1\)
**(b) Standard Gamma Density Curves Diagram**
This graph shows standard gamma density curves with different \(\alpha\) values:
- \(\alpha = 1\)
- \(\alpha = 0.6\)
- \(\alpha = 2\)
- \(\alpha = 5\)
**Flood Volume Modeling**
A study uses this distribution to model \(X = \) 3-day flood volume (\(10^8\) m\(^3\)) with the parameters \(\alpha = 11\), \(\beta = 8\), \(\gamma = 41\).
### Calculations
**(a) Mean and Standard Deviation of \(X\):**
- Mean: \(129 \times 10^8\) m\(^3\)
- Standard Deviation: \(26.5330 \times 10^8\) m\(^3\)
**(b) Probability of Flood Volume**
The probability that the flood volume is between 100 and 154 is \(0.281\).](https://content.bartleby.com/qna-images/question/f76fb667-0622-451e-86fc-d3866765f8e5/d94af18c-e6fb-48be-8d32-8b03ee6eec85/3ssnzmh_processed.png)
Transcribed Image Text:**Gamma Distribution with Threshold Parameter**
The two-parameter gamma distribution can be generalized by introducing a third parameter \(\gamma\), called a threshold or location parameter. This modifies the function as shown below:
\[
f(x; \alpha) =
\begin{cases}
\frac{x^{\alpha-1}e^{-x}}{\Gamma(\alpha)} & x \ge 0 \\
0 & \text{otherwise}
\end{cases}
\]
This adjustment amounts to shifting the density curves so they begin their ascent or descent at \(\gamma\) rather than 0.
**(a) Gamma Density Curves Diagram**
This graph illustrates several gamma density curves with different parameters \(\alpha\) and \(\beta\):
- \(\alpha = 2, \beta = \frac{1}{3}\)
- \(\alpha = 1, \beta = 1\)
- \(\alpha = 2, \beta = 1\)
**(b) Standard Gamma Density Curves Diagram**
This graph shows standard gamma density curves with different \(\alpha\) values:
- \(\alpha = 1\)
- \(\alpha = 0.6\)
- \(\alpha = 2\)
- \(\alpha = 5\)
**Flood Volume Modeling**
A study uses this distribution to model \(X = \) 3-day flood volume (\(10^8\) m\(^3\)) with the parameters \(\alpha = 11\), \(\beta = 8\), \(\gamma = 41\).
### Calculations
**(a) Mean and Standard Deviation of \(X\):**
- Mean: \(129 \times 10^8\) m\(^3\)
- Standard Deviation: \(26.5330 \times 10^8\) m\(^3\)
**(b) Probability of Flood Volume**
The probability that the flood volume is between 100 and 154 is \(0.281\).
![The two-parameter gamma distribution can be generalized by introducing a third parameter \(\gamma\), called a threshold or location parameter: replace \(x\) in
\[ f(x; \alpha) =
\begin{cases}
\frac{x^{\alpha-1}e^{-x}}{\Gamma(\alpha)} & x \geq 0 \\
0 & \text{otherwise}
\end{cases}
\]
by \(x - \gamma\) and \(x \geq 0\) by \(x \geq \gamma\). This amounts to shifting the density curves in the figure below so that they begin their ascent or descent at \(\gamma\) rather than 0.
**Figure Explanation:**
(a) **Gamma Density Curves**: This graph shows gamma density curves for different values of \(\alpha\) and \(\beta\):
- \(\alpha = 2\), \(\beta = \frac{1}{3}\)
- \(\alpha = 1\), \(\beta = 1\)
- \(\alpha = 2\), \(\beta = 1\)
(b) **Standard Gamma Density Curves**: This graph illustrates standard gamma density curves for various values of \(\alpha\):
- \(\alpha = 1\)
- \(\alpha = 0.6\)
- \(\alpha = 2\)
- \(\alpha = 5\)
A study employs this distribution to model \(X =\) 3-day flood volume \((10^8 \text{ m}^3)\). Suppose that values of the parameters are \(\alpha = 11\), \(\beta = 8\), \(\gamma = 41\) (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.)**
- **Mean**: \(129 \times 10^8 \text{ m}^3\)
- **Standard Deviation**: \(26.5330 \times 10^8 \text{ m}^3\)
**(b) What is the probability that flood volume is between 100 and 154? (Round your answer to three decimal places.)**
- **Probability**](https://content.bartleby.com/qna-images/question/f76fb667-0622-451e-86fc-d3866765f8e5/d94af18c-e6fb-48be-8d32-8b03ee6eec85/gj4dhk3_processed.png)
Transcribed Image Text:The two-parameter gamma distribution can be generalized by introducing a third parameter \(\gamma\), called a threshold or location parameter: replace \(x\) in
\[ f(x; \alpha) =
\begin{cases}
\frac{x^{\alpha-1}e^{-x}}{\Gamma(\alpha)} & x \geq 0 \\
0 & \text{otherwise}
\end{cases}
\]
by \(x - \gamma\) and \(x \geq 0\) by \(x \geq \gamma\). This amounts to shifting the density curves in the figure below so that they begin their ascent or descent at \(\gamma\) rather than 0.
**Figure Explanation:**
(a) **Gamma Density Curves**: This graph shows gamma density curves for different values of \(\alpha\) and \(\beta\):
- \(\alpha = 2\), \(\beta = \frac{1}{3}\)
- \(\alpha = 1\), \(\beta = 1\)
- \(\alpha = 2\), \(\beta = 1\)
(b) **Standard Gamma Density Curves**: This graph illustrates standard gamma density curves for various values of \(\alpha\):
- \(\alpha = 1\)
- \(\alpha = 0.6\)
- \(\alpha = 2\)
- \(\alpha = 5\)
A study employs this distribution to model \(X =\) 3-day flood volume \((10^8 \text{ m}^3)\). Suppose that values of the parameters are \(\alpha = 11\), \(\beta = 8\), \(\gamma = 41\) (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.)**
- **Mean**: \(129 \times 10^8 \text{ m}^3\)
- **Standard Deviation**: \(26.5330 \times 10^8 \text{ m}^3\)
**(b) What is the probability that flood volume is between 100 and 154? (Round your answer to three decimal places.)**
- **Probability**
Solution
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