Let $(X_1, X_2,..., X_n)$ be a vector of independent exponential random variables, say with density $f(x_1,x_2,...,x_n) = e^{-x_1}e^{-x_2}\cdots e^{-x_n}$ over $\mathbb{R}^n_+$. Is there a way to calculate (or upper bound) probabilities of the form $$ \mathbb{P}(q(\mathbf{X}) \geq 0)$$ where $q$ is an arbitrary quadratic form ?
Example : $ \mathbb{P}(X_2^2 + X_3^2 + 2X_2X_3 - X_1X_3 \geq 0)$ where $(X_1, X_2, X_3) \sim \mathcal{E}(1)^{\otimes 3}$