Expression defined by exponential random variables, probability of being nonnegative Consider $n \geq 2$. Let $E_1,...,E_n,F_1,...,F_n$ be independent exponentially distributed random variables with rate $1$. Define $T_E = \displaystyle \sum_{i=1}^{n}{E_i}$, and $T_F = \displaystyle \sum_{j=1}^{n}{F_j}$. 
Consider two arbitrary vectors $u$ and $v$ of $\mathbb{R}^n$, and define
$G_{u,v} = \displaystyle \sum_{i=1}^{n} \sum_{j=1}^{n}{u_i v_j (\dfrac{E_i}{T_E}-\dfrac{F_i}{T_F})(\dfrac{E_j}{T_E}-\dfrac{F_j}{T_F})}$
Question: is it possible to compute $\mathbb{P}(G_{u,v} \geq 0)$ as a function of the vectors $u$ and $v$? 
I tried to compute the multidimensional integral, but it gets very messy. I also have another formulation of my problem. The vectors $\{\dfrac{E_i}{T_E}\}$ and $\{\dfrac{F_i}{T_F}\}$ constructed above are random vectors of the n-dimensional simplex (with the uniform Dirichlet distribution). By rearranging the expression above, one can show that my problem is equivalent to finding the probability that $u.(p-q)$ and $v.(p-q)$ have the same sign, when $p$ and $q$ are independent random vectors of the simplex, and $u$ and $v$ are arbitrary vectors of $\mathbb{R}^n$.
I tried to rearrange these conditions, but did not make any progress. I also made some numerical simulations to see whether I could guess a simple relationship (for instance depending on $u.v$), in vain. Any ideas?
Thank you!

For each $a=(a_1,\ldots,a_n)$ define $G_a=\sum_{1\le i\le n}a_i\left(\frac{E_i}{T_E}-\frac{F_i}{T_F}\right)$. Then we have the identity $G_{u,v}=G_uG_v$. At this point $$\mathrm{Pr}(G_{u,v}\ge 0)=\mathrm{Pr}(G_u\ge 0)(G_v \ge 0)+\mathrm{Pr}(G_u\le 0)(G_v \le 0).$$
It implies that an answer to this question reduces to find the probability $\mathrm{Pr}(G_u\ge 0)$ for a given $u=(u_1,\ldots,u_n)$. From now on, it is just computations, unless there is a another quicker way :)
