Understanding the Gamma and Inverse Gamma Distributions: Definitions, Properties, and Connections

Gamma Distribution

Definition

If a random variable X has the probability density function

then X is said to follow a Gamma distribution, denoted X ~ Gamma(α, β).

Expectation and Variance

The expectation and variance are derived using integration by parts and properties of the Gamma function.

Special Cases

Various special cases arise when parameters take particular values, simplifying the distribution.

Relationship with Poisson and Exponential Distributions

If the number of events A occurring in a time interval follows a Poisson distribution with parameter λ, then the waiting time until the first event follows an exponential distribution with parameter λ.

The time interval between successive events follows an exponential distribution.

The waiting time for the k‑th event follows a Gamma distribution with shape k and rate λ.

Inverse Gamma Distribution

Definition

If a random variable X has the probability density function

then X is said to follow an inverse Gamma distribution, denoted X ~ InvGamma(α, β).

Expectation and Variance

Formulas for expectation and variance are provided analogously to the Gamma case.

Special Cases

Specific parameter choices lead to simplified forms of the inverse Gamma distribution.

References

https://blog.csdn.net/qq_42324085/article/details/118682965

https://www.its203.com/article/weixin_41875052/79843374