# Generalized exponential family of distributions

• johnaphun
In summary, the probability distribution for Y can be expressed as a member of the generalized exponential family of distributions, with specific values for the parameters θ, b, a, and ∅.
johnaphun

## Homework Statement

A discrete random variable Y has probability distribution given by

f(y;β) = (ky2β(y+k))/((β+3)(y+2k)(y+1)1/2)

## Homework Equations

I know that for a pdf to be from generalised exponential family of distribution it can expressed as

f(y) = exp{(yθ-bθ)/a∅ +c(y,∅)}

## The Attempt at a Solution

From exp{log(f(y;β))} i got -----> exp[(y+k)log(βy)-(y+2k)log(β+3)+log ky - (1/2)log(y+1)]

Is this sufficient or does it require further work?

Your attempt at a solution is a good start, but there are a few things that could be improved upon. First, you should specify that β > 0, as this is a requirement for the probability distribution to be well-defined. Additionally, you can simplify the expression by combining terms and using the properties of logarithms. The final expression should be of the form:

f(y;β) = exp{(yθ-bθ)/a∅ +c(y,∅)}

where θ = log(β), b = log(β+3), a = 1/2, ∅ = log(k), and c(y,∅) = (y+2k)log(β+3) - (y+k)log(βy) + log ky.

Overall, your approach is correct, but it would be helpful to be more explicit and organized in your notation.

## What is the Generalized Exponential Family of Distributions?

The Generalized Exponential Family of Distributions is a set of probability distributions that are defined by a specific mathematical form. These distributions are characterized by a set of parameters that determine the shape and behavior of the distribution.

## What are the key properties of the Generalized Exponential Family of Distributions?

The key properties of the Generalized Exponential Family of Distributions include a finite number of parameters, a natural parameterization, and a common support. Additionally, the distributions in this family are often closed under transformations and have conjugate prior distributions.

## What are some common examples of distributions in the Generalized Exponential Family?

Some common examples of distributions in the Generalized Exponential Family include the Gaussian (normal) distribution, the Poisson distribution, and the Bernoulli distribution. Other examples include the exponential, gamma, and beta distributions.

## What are the practical applications of the Generalized Exponential Family of Distributions?

The Generalized Exponential Family of Distributions has many practical applications in statistics, machine learning, and other fields. These distributions are often used to model real-world data and make predictions. They are also useful for parameter estimation and hypothesis testing.

## What are the main differences between the Generalized Exponential Family and the Exponential Family of Distributions?

The main difference between the Generalized Exponential Family and the Exponential Family of Distributions is that the former includes distributions with a finite number of parameters, while the latter includes distributions with a fixed number of parameters. Additionally, the Generalized Exponential Family is a superset of the Exponential Family, meaning that all distributions in the Exponential Family are also in the Generalized Exponential Family.

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