# Feasible to find the PDF of a chain of products of derived random variables?

I am trying to model the effect of (camera) pose error probabilistically on the position of a particular point. I don't believe that the specific details are directly relevant to my question, but I present here the actual expression to give an idea of its complexity.

Assuming that the vector $(x, y, z, \theta, \phi, \psi)$ represents the error "offset" in each of the six degrees of freedom of the pose, the point in question can be obtained from a pair of angles $\alpha_h$ and $\alpha_v$ along with a depth $z_0$ as follows:

$\mathbf{p}(\alpha_h, \alpha_v, z_0) = \left[ \begin{array}{c} (z_0 - z)\tan\left(\tan^{-1}\left( m\cos\left(\cos^{-1}\frac{\tan\alpha_h}{m} + \psi\right)\right) + \phi\right) + x \\ (z_0 - z)\tan\left(\tan^{-1}\left( m\cos\left(\cos^{-1}\frac{\tan\alpha_v}{m} + \psi\right)\right) + \theta\right) + y \\ z_0 \end{array} \right]$

where

$m = \sqrt{\tan^2 \alpha_h + \tan^2 \alpha_v}$

If $(x, y, z, \theta, \phi, \psi)$ are independent random variables, each with a known PDF (e.g. a zero-mean Gaussian process), is it feasible to compute an overall PDF for, say, the $x$-coordinate of $\mathbf{p}$? I get the impression that this is a chain of products of derived random variables, but I am not sure how to approach it.

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Chances of obtaining an analytic form are low. It is probably easiest to run simulations, by sampling $(x,y,z,\theta,\phi,\psi)$, applying the formula, and then using probability density estimator, such as either histogram, or smooth kernel distribution. This can be done in Mathematica, for example, using RandomVariate, TransformDistribution and SmoothKernelDistribution, for example. Let me know if you are interested in pursuing this direction. –  Sasha Jan 23 '12 at 17:59
@Sasha I suspected as much, and as a matter of fact my alternative idea was just such a Monte Carlo approach (and I am glad to hear someone who knows what they're talking about propose it!). I hadn't looked into the details of estimating the PDF from the data -- before reading your comment, I wasn't familiar with density estimation at all. I can certainly sample appropriate data in a closed-loop simulation. Would it be possible to parameterize the estimated PDF based on parameters like $\alpha_h$ and $\alpha_v$ so that it isn't necessary to simulate each new case? –  ezod Jan 23 '12 at 18:58
It depends very much of the nature of distributions for $(x,y,z,\theta,\phi,\psi)$. In case the distribution of $x$-coordinate of $\mathbf{p}$ is adequately represented by normal, or some other distribution with few parameters, across the range of $\alpha_v$ and $\alpha_h$, you may try to approximate the empirically measure dependence of those parameters on $\alpha_v$ and $\alpha_h$. This is tedious, but you owe it to yourself to try it. It might lead to one those "Eureka!" moments. –  Sasha Jan 23 '12 at 19:48