I have a set of data that I would like to simulate via a Dirichlet distribution, because I need that the sum is 1. I know that:
$$\Large E(x)=0.5151, Var(x)=0.086^2 E(y)=0.439, Var(y)=0.053^2 E(z)=0.045, Var(z)=0.014^2$$
I am trying to find the vector (a1,a2,a3) in the way that, if I sample in Dir(a1,a2,a3) then my sample set will have the characteristics previously mentionned.
Knowing the expenctancy and variance of the Dirichlet distribution variance and expectancy , I am building an equation system: \begin{cases}\LARGE 0.5151=\frac{a1}{(a1+a2+a3)} \\\LARGE 0.439=\frac{a2}{(a1+a2+a3)} \\\LARGE 0.045=\frac{a3}{(a1+a2+a3)}\\\LARGE0.086*0.086=\frac{a1*(a2+a3)}{((a1+a2+a3)*(a1+a2+a3)*(a1+a2+a3+1))}\\\LARGE0.053*0.053=\frac{a2*(a1+a3)}{((a1+a2+a3)*(a1+a2+a3)*(a1+a2+a3+1))}\\\LARGE 0.014*0.014=\frac{a3*(a1+a2)}{((a1+a2+a3)*(a1+a2+a3)*(a1+a2+a3+1))} \end{cases}
However, this system has 6 equations and 3 unknowns. Do you have any idea of how to solve it? Also, am I doing it correctly if I want to have a Dirichlet distribution to model my process?