# Distribution of Power Spectral Density in white noise

I have a signal $$s$$, of which the PSD I call as $$S$$.

So,

$$S(\omega) = \frac{1}{N} \left|\sum_{n = 0}^{N-1} s(n) \exp(-j\omega n)\right|^2$$

I have a closed form expectation of the above mentioned $$S$$:

$$\mathbb{E}[S(\omega)] = F(\Theta)$$ where $$F$$ is the expectation with some parameters $$\Theta$$.

However, the signal is corrupted with a white Gaussian noise of variance $$\sigma_n^2$$

$$z = s + n$$, where $$n \sim \mathcal{CN}(0, \sigma_n^2)$$.

I want to write the probability distribution of $$p(\Theta | Z )$$ , if the PSD of $$z$$ is $$Z$$.

I know that the PSD is exponentially distributed at each frequency point. So, in principle, $$S$$ is distributed exponentially with expectation $$F$$ and similarly, the noise is also exponentially distributed. I have used a probability that I write below, however, it gives me a big variance in my estimates of $$\Theta$$, although the bias is good enough.

$$\log(p(\Theta | Z )) = -\sum_{\omega} \left( L \log(F + \sigma_n^2) + \frac{\sum_{l= 1}^{L} Z_l(\omega)}{F + \sigma_n^2} \right)$$

where $$Z_l$$ are just different realizations of $$Z$$. Am I doing something wrong?