Proving a function is part of exponential family

Question

I have to prove this is in the exponential family

$P(X=x)=\frac{f(x)n^x}{g(n)}$ where $g(n)=\sum^\infty_{x=0}f(x)n^x$

so I have it split up as $I(x)_{0,1,2,3...}f(x)=h(x)$

$exp\big(\frac{xlog(n)}{(f(1)n+f(2)n^2+f(3)n^3...)}\big)=exp\big(\ x*\big(\frac{log(n)}{(f(1)n+f(2)n^2+f(3)n^3...)}\big)\big)$

can I call everything to the right of x inside the exponential $w_1(n)$ or are they each their own w(n) function.

Answer 1

Note that

$$\sum_{k=0}^{\infty} f(k) n^k = \sum_{x=0}^{\infty} f(x) n^x$$

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A distribution $g(x;n)$ belongs to an exponential family if it can be represented in the form:

$$g(x;n)=\nu(x)\exp[\xi(n)T(x)-S(n)]$$

We have

$$g(x;n) = P(X = x; n) = f(x) \exp[\ln(n) x -\ln[\sum_{k=0}^{\infty} f(k) n^k]]$$