# Posterior Distribution of Beta Prior

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.

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

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