P.D.F of a norm of sheared Gaussian vector

**Question:** Let $\displaystyle W_i \sim {\cal {N}}(0,\sigma ^2)$ for $\displaystyle i = 1,2,3,...,n$. For $\displaystyle (c_1,c_2,...,c_n) \in \mathbb{R}^n$, define $\displaystyle Y = \sum_{i = 1}^{i = n} c_i W_i ^2$. **Compute **$\displaystyle \text{Pr}\{Y \ge 0\}$

(Note that the constants $\displaystyle c_i$ can be negative,thus its not exactly non-central chi square)

**My Attempt:** I tried computing the pdf. I tried to use the M.G.F trick. We know that $\displaystyle M_{W_i ^2}(s) = \frac1{\sqrt{1 + 2s}}$. Clearly $\displaystyle M_Y(s) = \prod_{i=1}^{i=n} M_{W_i ^2}(sc_i) = \prod_{i=1}^{i=n}\frac1{\sqrt{1 + 2sc_i}}$

I am stuck here. I have forgotten Fourier Transforms, I think :(

Regards,

Srikanth