Multiparameter exponential family

**serious331** · May 20th 2010, 07:51 AM

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

**harish21** · May 20th 2010, 07:54 AM

Originally Posted by serious331

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

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

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

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

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