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Explain the two parameterizations of the gamma distribution in common use.
Explain the two parameterizations of the gamma distribution in common use.
The two parameterizations of the gamma distribution in common use are: 1) With a shape parameter $k$ and a scale parameter $\theta$, and 2) With a shape parameter $\alpha = k$ and an inverse scale parameter $\beta = 1 / \theta$, called a rate parameter.
What are the constraints on the parameters of the gamma distribution?
What are the constraints on the parameters of the gamma distribution?
In both parameterizations, the shape parameter and the scale parameters are positive real numbers.
What is the maximum entropy probability distribution for a random variable X in the gamma distribution?
What is the maximum entropy probability distribution for a random variable X in the gamma distribution?
The gamma distribution is the maximum entropy probability distribution for a random variable $X$ for which $E[X] = k\theta = \alpha/\beta$ is fixed and greater than zero, and $E[\ln(X)] = \psi(k) + \ln(\theta) = \psi(\alpha) - \ln(\beta)$ is fixed (where $\psi$ is the digamma function).
What are the special cases of the gamma distribution?
What are the special cases of the gamma distribution?
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What is the relationship between the shape parameter $k$ and the rate parameter $\beta$?
What is the relationship between the shape parameter $k$ and the rate parameter $\beta$?
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Study Notes
Gamma Distribution Overview
- The gamma distribution is a two-parameter family of continuous probability distributions in probability theory and statistics.
- Special cases of the gamma distribution include the exponential distribution, Erlang distribution, and chi-squared distribution.
Parameterization
- Two common parameterizations:
- Shape parameter k and scale parameter θ.
- Shape parameter α = k and inverse scale parameter β = 1 / θ, known as the rate parameter.
- Both parameters are positive real numbers in each form.
Maximum Entropy
- The gamma distribution represents the maximum entropy probability distribution for a random variable X with the following fixed conditions:
- Expected value E[X] = kθ = α/β is greater than zero.
- The expected value of the natural logarithm E[ln(X)] = ψ(k) + ln(θ) = ψ(α) − ln(β) is also fixed, where ψ denotes the digamma function.
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Description
Test your knowledge of the gamma distribution with this quiz. Explore the two common parameterizations, special cases, and applications in probability theory and statistics.