Assume I am looking for a normal distribution $\mathcal{N}(\mu,\Sigma)$. For simplicity let's say we only have 2 random variables $x$ and $y$ and a known $\mu=0$.
Is it possible to estimate $\Sigma$ by observing the variance along multiple directions?
For example, I measure the variance $\sigma_1$ along the vector $\mathbb{v}_1 = (x_1,y_1)^T$. In another step I obtain a different variance $\sigma_2$ from a different direction $\mathbb{v}_2 = (x_2,y_2)^T$. Ideally one would continue to observe these single variations in different directions and combine them in one multivariate normal distribution.
Does this make sense?
EDIT: Some additional background information might be useful: I have a sensor device with known position and orientation in 2D space (in a future step both may have an uncertainty). The sensor is able to measure only the distance of a point along its orientation. I'm also given the sensor model. So for each distance measure $d_i$, I obtain the standard error $\sigma(d_i)$ which depends on the distance.
Since I'm not able to manipulate the sensor position to my advantage or perform a large number of measurements, I'd like to combine these variances into one covariance matrix in order to make a more reliable prediction of the position of the measured point.
This is just a thought that is still under development with no guaranty to work out correctly. Hence my question of "making sense"...