I was attempting to calculate a mathematical expectation since yesterday. It seems complicate for me and i can not do it. May be someone can help me here.
Let be $u_1$ and $u_2$ two positions on a 3 dimensional unit hypersphere. I need to calculate $$P=\int_{z}\frac{\exp(\alpha_1 + \beta u_1z)}{\exp(\alpha_1 + \beta u_1z)+\exp(\alpha_2 + \beta u_2z)}dF(z)$$ Suppose that $z \sim \mathcal{M}_3(\eta,v)$, that is a 3 dimensional von Mises-Fisher distribution with intensity parameter $\eta$ and mean directional parameter $v$.
In fact the answer can help me to compute for the general case which is $$P_i=\int_z\dfrac{\exp(\alpha_i + \beta u_iz)}{\sum_{i=1}^{n} \exp(\alpha_i + \beta u_iz)}dF(z)$$ with $\alpha_i$, $\beta$ $\in$ $\mathbb{R}$. Thanks!