Suppose we have a network with node $i$ in the origin $[0,0]$ and node $1$ and node $2$ located in $[x_1,y_1]$ and $[x_2,y_2]$ respectively, where $x_1,y_1,x_2,y_2$ follow a Uniform distribution $\sim\mathcal{U}(-L,L)$ . However, we also have estimated positions so for each node, ${[\hat{x}_i,\hat{y}_i]}=[x_i,y_i]+\vec{e}_i \sim \mathcal{N}(0,\sigma^2\mathbf{I}_{2 \times 2})$. This means that we add a gaussian noise to the real locations to have the estimated positions, based on the variance $\sigma^2$ we define.
We want to find $P[\hat{d}_{i1}<\hat{d}_{i2} \mid d_{i1}<d_{i2}]$ where $\hat{d}_{i1}$ is the estimated distance from point 1 to origin, $\hat{d}_{i2}$ is the estimated distance from point 2 to origin, $d_{i1}$ is the real distance from point 1 to origin and $d_{i2}$ is the real distance from point 2 to origin using the real positions or estimated ones accordingly, so:
$d_{i1}=\sqrt{(x_1)^2+(y_1)^2}$
$d_{i2}=\sqrt{(x_2)^2+(y_2)^2}$
$\hat{d}_{i1}=\sqrt{(\hat{x}_1)^2+(\hat{y}_1)^2}$
$\hat{d}_{i2}=\sqrt{(\hat{x}_2)^2+(\hat{y}_2)^2}$
The answer has to be based on the variance of the noise $\sigma^2$ so depending on that value, the probability will be higher or lower.
How would you approach this problem?
We know that $P[d_{i1}<d_{i2}]=1/2$ and $P[\hat{d}_{i1}<\hat{d}_{i2}]=1/2$ for random variables with the same parameters, but the conditional probability is causing me problems.
Then, how would you find $P[\hat{d}_{i1}<\hat{d}_{i2} \mid d_{i1}<d_{i2}]$?