# Thread: Bayesian - Exponential Family

1. ## Bayesian - Exponential Family

Im trying to show that the exponential distribution is a member of the exponential family. I can only find examples of normal, binomial and bernoulli as guidance on the web. I know that exponential is conjugate to gamma. Not really sure of the exact process. Please can someone point me in the right direction? Thanks.

2. You just have to factor it into the following form:

$\displaystyle f(x|\beta) = h(x)c(\beta)e^{\sum_{i = 1} ^ k t_i(x)w_i(\beta)}$

for some functions h, c, t_i, w_i, and positive integer k. They can be constant functions if you want. For the exponential density, I'll get you started by giving you

$\displaystyle f(x|\beta) = \frac{1}{\beta}e^{-x/\beta}I_{(0, \infty)}(x)$ (just so there's no ambiguity, since some people write it differently)
$\displaystyle h(x) = I_{(0, \infty)}(x)$
$\displaystyle c(\beta) = \frac{1}{\beta}$