A Gamma random variable has mean of 5.4 and variance of 16.2. Find the parameters of this distribution.
A Gamma random variable has mean of 5.4 and variance of 16.2. Find the parameters of this distribution.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![**Problem Statement:**
A Gamma random variable has a mean of 5.4 and variance of 16.2. Find the parameters of this distribution.
**Given Solutions:**
- \(\alpha = 16.20\) (Incorrect)
- \(\beta = 0.3333\) (Incorrect)
**Explanation:**
The Gamma distribution is characterized by two parameters, usually denoted as \(\alpha\) (shape) and \(\beta\) (rate). The mean \(\mu\) and variance \(\sigma^2\) for a Gamma distribution are expressed as:
\[
\mu = \frac{\alpha}{\beta}
\]
\[
\sigma^2 = \frac{\alpha}{\beta^2}
\]
Using the given values:
- Mean (\(\mu\)) = 5.4
- Variance (\(\sigma^2\)) = 16.2
To solve for \(\alpha\) and \(\beta\), two equations are derived from the expressions:
1. \(\frac{\alpha}{\beta} = 5.4\)
2. \(\frac{\alpha}{\beta^2} = 16.2\)
By solving these equations simultaneously, you can determine the correct parameters \(\alpha\) and \(\beta\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d7ec503-389e-4b40-b2fe-31c0ac723423%2F556bb4d9-4968-4c76-81ee-555fc3f2a40b%2Fng96nfl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A Gamma random variable has a mean of 5.4 and variance of 16.2. Find the parameters of this distribution.
**Given Solutions:**
- \(\alpha = 16.20\) (Incorrect)
- \(\beta = 0.3333\) (Incorrect)
**Explanation:**
The Gamma distribution is characterized by two parameters, usually denoted as \(\alpha\) (shape) and \(\beta\) (rate). The mean \(\mu\) and variance \(\sigma^2\) for a Gamma distribution are expressed as:
\[
\mu = \frac{\alpha}{\beta}
\]
\[
\sigma^2 = \frac{\alpha}{\beta^2}
\]
Using the given values:
- Mean (\(\mu\)) = 5.4
- Variance (\(\sigma^2\)) = 16.2
To solve for \(\alpha\) and \(\beta\), two equations are derived from the expressions:
1. \(\frac{\alpha}{\beta} = 5.4\)
2. \(\frac{\alpha}{\beta^2} = 16.2\)
By solving these equations simultaneously, you can determine the correct parameters \(\alpha\) and \(\beta\).
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