# Bayes: posterior density, mean and mode [duplicate]

The likelihood function is

$$f(n_1|\alpha,n) = [\frac{n!}{n_1!(n-n_1)!}]\alpha^{n_1}(1-\alpha)^{n-n_1}$$

The prior for $$\alpha$$

$$g(\alpha)=(\alpha(1-\alpha))^{-1}$$ for $$\alpha$$ between 0 and 1.

I know that $$f(n_1|\alpha,n)g(\alpha)$$ is the posterior density. But how can i find the correct form? I suspect it must be the pdf of a beta-distribution.

And how do I calculate the mean and mode afterwards?