I'm doing a project on random matrices and its applications. I have the joint probability density and want to calculate the probability of $s=\sum\_{j=1}^N\lambda\_j^2$. So we have


$$P(s)=C\_{N,K}\int...\int\delta\left(s-\sum\_{j=1}^N\lambda\_j^2\right)\delta\left(1-\sum\_{j=1}^N\lambda\_j\right)\prod\_i\lambda\_i^{K-N}\prod\_{i<j}(\lambda\_i-\lambda\_j)^2\prod\_id\lambda\_j$$


The limits being from $-\infty$ to $\infty$ for all the integrals and $N,K$ are constants. Now I tried expanding the Vandermonde determinant after getting rid of the delta functions and tried to use Selberg integral but unfortunately this is turning out to be very tedious and cumbersome. Even if its possible we'll end up sums of functions of gamma functions. Is there any other way to do this?



Probability of a quantity from Ginibre ensemble
The expectation value of $s=\sum\_{j}\lambda\_j^2$ follows from equation 5.11 of [arXiv:quant-ph/0405031](https://arxiv.org/abs/quant-ph/0405031),
$$E(s)=\frac{K+N}{KN+1}.$$
The second moment is given by equation 5.16, and it is already a very lengthy expression.


