Let me preface this by saying that I'm an engineer, and by no means a mathematician, so please excuse any mathematical "wrong-doing" in my explanation.
I have two vectors $V_1$ and $V_2$, whose coordinates in cartesian space $(X_1, Y_1)$ and $(X_2, Y_2)$ are defined as follows :
\begin{cases} X_i = R_i^1*cos(\alpha_i) + R_i^2*cos(\beta_i) \\[2ex] Y_i = R_i^1*sin(\alpha_i) + R_i^2*sin(\beta_i) \end{cases}
where $\alpha_i$ are known constants, $R_i^j$ are random variables with known mean and variance (and perhaps normal distribution, but I am not certain it applies to all my cases), and $\beta_i$ are random variables with uniform distribution: basically my vectors are composed of a variable distance in a known direction given by angle $\alpha_i$ + a variable distance in an unknown direction given by angle $\beta_i$. It is very easy to derive expected value and variance of $X_i$ and $Y_i$ because everything is known, which allows me to work in the cartesian space.
What I'm aiming to do is determine the expected value and variance of the norm $D$ of the difference between the two vectors $V_1$ and $V_2$:
$D = | V1 - V2 | = \sqrt{(X_1 - X_2)^2 + (Y_1 - Y_2)^2} = \sqrt{\Delta X^2 + \Delta Y^2}$
I know the expected value and variance of $X_1$, $Y_1$, $X_2$ and $Y_2$ and therefore am able to approximate mean and variance of $\Delta X^2$ and $\Delta Y^2$ via the law of the unconscious statistician and a second order Taylor series (because the distribution is not known). For example with $\Delta X^2$ :
$E(\Delta X) = E(X_1) - E(X_2)$ (where these values are known)
$var(\Delta X) = var(X_1) + var(X_2)$ (where these values are known)
$E(\Delta X^2) \approx E(\Delta X)^2 + var(\Delta X)^2$
$var(\Delta X^2) \approx 4*E(\Delta X)^2*var(\Delta X)^2$
I check the accuracy of this approximation through random sampling and it is accurate enough for my needs.
Now, for the step on which I am stuck, I need to compute the expected value and variance of $\Delta X^2 + \Delta Y^2$. This amounts, through the covariance formulae, to knowing the expected value of $\Delta X^2 \Delta Y^2$:
$var(\Delta X^2+\Delta Y^2) = var(\Delta X^2) + var(\Delta Y^2) + 2cov(\Delta X^2,\Delta Y^2) = var(\Delta X^2) + var(\Delta Y^2) + 2E(\Delta X^2 \Delta Y^2) - 2E(\Delta X^2)E(\Delta Y^2)$
The problem being that $(X_i, Y_i)$ are dependent variables, and therefore so are $\Delta X^2$ and $\Delta Y^2$.
Through my research I have found two similar questions with interesting answers (n°1 and n°2), but I seem to be stuck because in one case $\sigma_{xy}$ is known (which is not my case), and in the other the correlation coefficient $\rho$ seems to be known, which is also (to my knowledge) not my case. In my experience (through random sampling), $\Delta X$ and $\Delta Y$ do seem to be jointly normal. I have therefore tried to use the formulae from the second post, using $\rho$ as a kind of potentiometer to fit the result to my sampling (which I guess is a natural approach to an engineer but may seem horrible to a mathematician), but this is not at all robust to even minor tweaks to the initial model ($R_i^j$, $\alpha_i$, etc).
Is there any other mathematical "trick" to figuring this out ?
Once I have $var(\Delta X^2 + \Delta Y^2)$ through $E(\Delta X^2 \Delta Y^2)$, I expect to be able to compute $var(D)$ and $E(D)$ using the law of the unconscious statistician and a Taylor series, but that is not really the subject of this post.
Thank you for your help, I hope all of this is clear enough.
