Question
If I have n independent random normal numbers denoted $X_i$ each with mean $\mu_i$ and variance $\sigma_i$ (for $i = 1 ... n$). For each $X_i$ I have a weighting factor $w_i$. What is the cumulative distribution function for the random variable $Z$ defined as: $$Z = \sum^n_{i=1} w_i \cdot X_i^{-2}$$
Motivation
To determine the likelihood of a point in space receiving no less than a given radiation exposure (kerma) given a series of radioactive point sources in the vicinity with an uncertainty in their position.
$X_i$ is distance between the point source and the calculation point in space. The $w_i$ factor is the time which the source spends at that distance. The resulting $Z$ is the total resulting radiation exposure (kerma) to that point in space.
My aim is to optimise over a grid of calculation points to minimise the likelihood that any point in space receives less than the defined desired exposure.
What I have tried
I have been looking into the the convolution of the inverse gamma distribution. It seems like $w_i\cdot X^{-2}_i$ might be distributed like that. However I have found myself rather stumped.
A link to a screenshot of what the CDF looks like is here: https://i.stack.imgur.com/0josm.png
I am wondering if I work out the actual PDF of $w_i\cdot X^{-2}_i$ then I might be able to do the fourier transform, take the product of the individual transformed PDFs then inverse transform to get the convolution. Then take the integral for the CDF. However to do this I need to work out what exactly is the PDF of $w_i\cdot X^{-2}_i$.
Any help would be greatly appreciated. This calculation is within the very middle of a global optimiser which requires it to run a very large number of times. If I can find an analytical solution it would be perfect.