Question

If X has a Poisson distribution and the prior distribution of its parameter Λ is a gamma distribution with the parameters α and β, show that the posterior distribution of Λ given X=x is a gamma distribution with the parameters α+x andβ/(β+1)

Answer 1

To prove the posterior distribution of Λ is a gamma distribution with parameters α+x and β/(β+1) given X=x from a Poisson distribution with Λ's prior as a gamma distribution, Bayes' theorem is applied, revealing the conjugacy relationship between Poisson and gamma distributions.

Step-by-step explanation:

The question is asking to show that if a random variable X has a Poisson distribution and the prior distribution of its parameter Λ is a gamma distribution with parameters α and β, then the posterior distribution of Λ given X=x is also a gamma distribution with new parameters α+x and β/(β+1). To show this, we begin by applying Bayes' theorem in the context of continuous distributions (specifically, the conjugacy relationship between the Poisson and gamma distributions).

The prior distribution of Λ is gamma(α, β). The likelihood function given that X=x is derived from the Poisson distribution of X. Using the properties of gamma and Poisson distributions, we can combine the prior and the likelihood to find the posterior. In the Bayesian framework, the posterior distribution is proportional to the product of the likelihood and the prior distribution:

P(Λ|x) ∑ P(x|Λ)P(Λ)

By substituting the respective probability density functions and simplifying, we find that the posterior distribution of Λ is indeed a gamma distribution with updated parameters α+x and β/(β+1), demonstrating the conjugacy between Poisson and gamma distributions in Bayesian inference.