When I have a simple Bernoulli trial with a certain variable taking, for instance, values 0 and 1, I have a constant prior distribution for the $\theta$ parameter, i.e. pdf $p(\theta) = 1$ between 0 and 1. That ends up meaning that the first moment of the distribution, which is the probability that I'll observe each value, is 1/2.
My question is, what would the equivalent pdf for $\theta$ be in the case where my variable can take three values, such as 0, 1 or 2? And what if it can take n values?
For every $n\geqslant2$, the space $\Delta_n$ of parameters when the random variable can take $n$ values is made of the tuples $(p_k)_{1\leqslant k\leqslant n}$ in $(\mathbb R_+)^n$ such that $\sum\limits_{k=1}^np_k=1$. The uniform measure on $\Delta_n$ is called a Dirichlet distribution, corresponding to the concentration parameters $(\alpha_k)_{1\leqslant k\leqslant n}$ all equal to $1$.
For $n=3$, the density with respect to Lebesgue measure on $\mathbb R^2$ where $(p_1,p_2)$ lives is $$ f(p_1,p_2)=2\cdot\mathbf 1_{p_1\gt0,p_2\gt0,p_1+p_2\lt1}. $$ For every $n\geqslant2$, the mean value of each coordinate of the tuple is $\frac1n$, if only by symmmetry. Likewise, the variance of each coordinate is $\frac{n-1}{n^2(n+1)}$ and the covariance of any two of them is $\frac{-1}{n^2(n+1)}$.