Question

For a gamma distribution with a mean of 4.0 mm and a variance of 3.7 mm, what are the shape (a) and rate (β) parameters?

1) Shape (a) = 6.944 and rate (β) = 0.625
2) Shape (a) = 2.8 and rate (β) = 3.6
3) Shape (a) = 6.000 and rate (β) = 0.666
4) Cannot be determined

Answer 1

The given mean and variance for the gamma distribution suggest a relationship between the shape (a) and rate (β) parameters, but the computations appear inconsistent with the provided answer options, possibly suggesting incorrect given values or a misunderstanding.

Step-by-step explanation:

The question pertains to the gamma distribution and involves finding the shape and rate parameters given the mean and variance. To find these parameters we can use the following relationships:

• The mean of a gamma distribution μ is given by μ = a/β, where a is the shape parameter and β is the rate parameter.
• The variance σ² is given by σ² = a/β².

Given a mean μ = 4.0 mm and variance σ² = 3.7 mm², we can solve for a and β.

Using the mean, we rearrange the first equation to find a = μβ. Substituting into the second equation, we get μ²β = 3.7, so β = 3.7/μ² = 3.7/16 = 0.23125. Now, using this value for β in a = μβ gives us a = 4 * 0.23125 = 0.925. However, these computations do not match the given options and may indicate a misunderstanding or that the values provided are incorrect.