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I have an equation for the probability generating function of a negative binomial $(n,p)$ distribution:

$$G(x)= \Biggl(\frac{px}{1-(1-p)x}\Biggr)^n$$

I'd like to solve for $x$ in the following equation:

$$x =\Biggl(\frac{px}{1-(1-p)x}\Biggr)^n$$

But i'm not sure how to solve it. I tried this:

$$x^{1-n}= \frac{p^n}{(1-(1-p)x)^n}$$ but that's all to where i got to. How do I solve for $x$ here?

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    Numerically solve for the roots perhaps – tomka Feb 25 '20 at 13:04
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    Isn't $x=0$ always a solution? Are you looking for some other solution? – josliber Feb 25 '20 at 14:14
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    I think $x=1$ is a solution. – Masoud Feb 25 '20 at 15:10
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    Your algebra is incorrect. When you clear the denominator, this is a polynomial equation of degree $n+1,$ so it has $n+1$ complex roots, of which at least one but as many as $n+1$ will be real. Unless $n$ is tiny, you need to solve it numerically. But could you explain what the statistical application of this solution would be? (One has to wonder whether you got to this point erroneously by trying to solve some other problem.) – whuber Feb 25 '20 at 15:52

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