I would like your opinion on the following problem which I'm trying to tackle.
A boat's radio can be heard from a distance and depending on this distance, the noise of the sensed signal varies. Let's assume that given a noise, the probability that the viewer lies at a specific distance to the boat can be described by the following Maxwell–Boltzmann distribution:
$p(d|n) = \sqrt{\frac{2}{\pi}}\frac{d^{2}e^{-\frac{d^{2}}{2n^{2}}}}{n^{3}}$, where d≥0
Let's assume now that a receiver detects 2 signals (from 2 independent boats). A signal with noise 1500 and a signal with noise 8000. These two corresponding PDFs suggest that the receiver should be around 2500 meters from the first boat and 12km from the second boat.
Taking each of the functions above and replacing $d$ with $\sqrt{(x-cx)^{2}+(y-cy)^{2}}$ (from the circle equation), I guess we get a new $f(x,y)$ function, where, given a CenterPoint[$cx, cy$], the PDF is now rotated about the Z-axis of the CenterPoint (like a "solid of revolution"). For example, assuming that I had a prior belief that the first boat was at the CenterPoint$[cx=30000,cy=30000]$ and the second boat was at the CenterPoint$[cx=15000,cy=15000]$, then the above PDF's would lead to these two revolutions.
Although I have a belief regarding the position of each boat, I have no idea of where the receiver is. Therefore, I thought of finding where the product of these functions becomes maximum, to infer that the receiver is most probably at that point.
*Update: I would like to mention that in this example, I am using only 2 boats. However, I might have signals from $n$ boats and thus, need to maximise the product of n functions.
With that said, my question is.. do I have first to normalize each f(x,y) function so that the volume beneath it is 1? I ask because in this example where the only difference between the two (top-left) PDF's is their location (and thus, $P_{A}(5)=P_{B}(10)$), yet, a rotation of them (which would lead to the bottom functions) would produce different normalization constants and so, I guess.. $P_{A}(0,5)≠P_{B}(0,10)$.
What are your thoughts/suggestions/corrections?
Thx for your time, if I had some reputation here I would place some bounty too.
If you are trying to find $(x,y)$ that maximizes the product
$$p(\sqrt{(x-x_A)^2 + (y -y_A)^2} \mid n_A) \times p(\sqrt{(x-x_B)^2 + (y-y_B)^2} \mid n_B)$$
then normalizing would not matter at all. I mean it literally would not change the result, i.e. the optimal $(x,y)$, because
$${p(\sqrt{(x-x_A)^2 + (y -y_A)^2} \mid n_A) \over K_A} \times {p(\sqrt{(x-x_B)^2 + (y-y_B)^2} \mid n_B) \over K_B}$$
has the same argmax $(x,y)$ regardless of the normalizing constants $K_A, K_B$.
There might be other ways of estimating $(x,y)$ where normalizing would change the result, but not when you're just maximizing the product of the densities (a sort of max-likelihood estimator).