Generalized exponential family of distributions

[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?

 

[mighty2000]

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.