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 x20 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,) 4 1.0-a=2, B =} 1.0- a = | -a = 1, ß = 1 05- a = .6 ‚a = 2, B = 2 0.5 - a =2 a = 5 .a = 2, ß = 1 i ż 3 2 3 4 3 2 3 6 7 (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 = 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 20.7846 v 10°m3 mean standard deviation 10®m3 (b) What is the probability that flood volume is between 100 and 155? (Round your answer to three decimal places.) 9.16 (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 OSth percentile of the flood volume distribution? (Round vour answer to two decimal places
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 x20 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,) 4 1.0-a=2, B =} 1.0- a = | -a = 1, ß = 1 05- a = .6 ‚a = 2, B = 2 0.5 - a =2 a = 5 .a = 2, ß = 1 i ż 3 2 3 4 3 2 3 6 7 (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 = 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 20.7846 v 10°m3 mean standard deviation 10®m3 (b) What is the probability that flood volume is between 100 and 155? (Round your answer to three decimal places.) 9.16 (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 OSth percentile of the flood volume distribution? (Round vour answer to two decimal places
A First Course in Probability (10th Edition)
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Chapter1: Combinatorial Analysis
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![**The Three-Parameter Gamma Distribution**
The two-parameter gamma distribution can be generalized by introducing a third parameter, \(\gamma\), called a threshold or location parameter. To do this, replace \(x\) in the equation:
\[
f(x; \alpha) =
\begin{cases}
\frac{x^{\alpha-1}e^{-x}}{\Gamma(\alpha)} & x \geq 0 \\
0 & \text{otherwise}
\end{cases}
\]
by \(x - \gamma\) with \(x \geq \gamma\). This adjustment amounts to shifting the density curves in the figure so that they begin their ascent or descent at \(\gamma\) rather than 0.
### Graphs and Explanation:
**(a) Gamma Density Curves:**
- The graph displays several gamma density curves with varying parameters:
- \(\alpha = 2, \beta = \frac{1}{3}\)
- \(\alpha = 1, \beta = 1\)
- \(\alpha = 2, \beta = 2\)
- \(\alpha = 2, \beta = 1\)
The x-axis represents the variable \(x\), and the y-axis represents the density \(f(x; \alpha, \beta)\). The curves show how the shape changes with different \(\alpha\) and \(\beta\) values.
**(b) Standard Gamma Density Curves:**
- The graph illustrates standard gamma density curves for:
- \(\alpha = 1\)
- \(\alpha = 0.6\)
- \(\alpha = 2\)
- \(\alpha = 5\)
These curves highlight the changes in the distribution's shape as \(\alpha\) varies, with all plotted against the same axes as in (a).
### Study Application:
A study uses this distribution to model \(X =\) 3-day flood volume \((10^8 \, \text{m}^3)\). Given parameter values are \(\alpha = 12\), \(\beta = 6\), and \(\gamma = 43\).
**(a) Mean and Standard Deviation:**
- Mean: \(115 \times 10^8 \, \text{m}^3\)
- Standard Deviation:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee8fb823-9e56-4d34-a863-64debae68570%2F1aabc86d-790a-4143-b3fb-a8fc1ced5366%2Fbw7lpp_processed.png&w=3840&q=75)
Transcribed Image Text:**The Three-Parameter Gamma Distribution**
The two-parameter gamma distribution can be generalized by introducing a third parameter, \(\gamma\), called a threshold or location parameter. To do this, replace \(x\) in the equation:
\[
f(x; \alpha) =
\begin{cases}
\frac{x^{\alpha-1}e^{-x}}{\Gamma(\alpha)} & x \geq 0 \\
0 & \text{otherwise}
\end{cases}
\]
by \(x - \gamma\) with \(x \geq \gamma\). This adjustment amounts to shifting the density curves in the figure so that they begin their ascent or descent at \(\gamma\) rather than 0.
### Graphs and Explanation:
**(a) Gamma Density Curves:**
- The graph displays several gamma density curves with varying parameters:
- \(\alpha = 2, \beta = \frac{1}{3}\)
- \(\alpha = 1, \beta = 1\)
- \(\alpha = 2, \beta = 2\)
- \(\alpha = 2, \beta = 1\)
The x-axis represents the variable \(x\), and the y-axis represents the density \(f(x; \alpha, \beta)\). The curves show how the shape changes with different \(\alpha\) and \(\beta\) values.
**(b) Standard Gamma Density Curves:**
- The graph illustrates standard gamma density curves for:
- \(\alpha = 1\)
- \(\alpha = 0.6\)
- \(\alpha = 2\)
- \(\alpha = 5\)
These curves highlight the changes in the distribution's shape as \(\alpha\) varies, with all plotted against the same axes as in (a).
### Study Application:
A study uses this distribution to model \(X =\) 3-day flood volume \((10^8 \, \text{m}^3)\). Given parameter values are \(\alpha = 12\), \(\beta = 6\), and \(\gamma = 43\).
**(a) Mean and Standard Deviation:**
- Mean: \(115 \times 10^8 \, \text{m}^3\)
- Standard Deviation:
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