Multiparameter exponential family

Feb 2010
37
0
Let X be distributed as \(\displaystyle N(\mu, \sigma^2)\) with n=2 and \(\displaystyle \theta = (\mu, \sigma) \in R \times R^{+}\), where mu and sigma are treated as parameters.. How should I show that this belongs to a two parameter exponential family?
 
Feb 2010
1,036
386
Dirty South
Let X be distributed as \(\displaystyle N(\mu, \sigma^2)\) with n=2 and \(\displaystyle \theta = (\mu, \sigma) \in R \times R^{+}\), where mu and sigma are treated as parameters.. How should I show that this belongs to a two parameter exponential family?

Hint: \(\displaystyle f(x; \theta)\) belongs to a two parameter exponential family if you can express


\(\displaystyle f(x; \theta) = a(\theta).g(x). \mbox{exp} ( \sum_{i=1}^{2} {b_{i}}(\theta). {R_{i}(x)} ) \)

Now try to express the pdf of your normal dist. in the above form
 
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