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I have three values ($x_1=2$, $x_2=8$, and $x_3=4$). These are drawn from a Binomial distribution with parameter $k=12$ and unknown parameter $p$. The prior distribution of $p$ is a Beta distribution with parameters $\alpha=3$ and $\beta=1$.

I know that the posterior would be a Beta distribution with parameters $\alpha+\sum x_i$ and $\beta+n-\sum x_i$.

So with that in mind, the answer would be $p \sim Beta(17,-10)$

But I'm confused on if:

1) I should be using the parameter $k$ somewhere or if it is not needed and

2) If a parameter can be negative or if I did something wrong.

I have very minimal knowledge of this subject and am just starting to learn.

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    $\begingroup$ How do you get the -10 for $\beta$? $\endgroup$ – Lerner Zhang Apr 15 '20 at 0:37
  • $\begingroup$ Prior β=1, n=3 and the sum =14. So 1+3-14=-10 $\endgroup$ – mmd123 Apr 15 '20 at 0:58
  • $\begingroup$ In beta distribution , both parameters must be positive. $\endgroup$ – Masoud Apr 15 '20 at 1:17
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If $p\sim Beta(\alpha , \beta) $ so

$$\Pi (p|x)\propto f(x_1,x_2,x_3|p) \Pi(p)=p^{\sum x_i + \alpha -1} (1-p)^{3n-\sum x_i+ \beta -1} $$ so

$$p|x \sim Beta(\alpha +\sum x_i=17 , 3n+\beta-\sum x_i=3*12+1-17=20).$$

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