PDF of distance between measured coordinates with normally distr. errors

[omg!]

hi,
the problem i have is this:

consider a line segment L of length d in R^3. the projection of the endpoints of L on a fixed plane can be determined with an error that is normally distributed, and the errors for the two endpoints needs not to be equal. Assuming that L is randomly oriented in R^3, what is the distribution of the projected distance?

I have been thinking about this for quite a while, reading up on chi(-squared), rice, reighley distributions. the problem is that the coordinates are not independent, so that dX,dY in sqrt(dX^2+dY^2) are not independent, making these distributions unsuitable. The PDF of the projected distance neglecting contribution of errors is

r/d*1/sqrt(d^2-r^2)

where r is the projected distance on the plane.

Thanks for all the help

 

[mmwave]

and suggestions!

Hi there,

Thank you for sharing your problem with us. This is an interesting question that requires some mathematical analysis to find the distribution of the projected distance.

First, let's define the problem more clearly. We have a line segment L of length d in R^3, and we want to find the distribution of the projected distance on a fixed plane. The endpoints of L have normally distributed errors, which means that the projected distance will also have a normal distribution. However, the errors for the two endpoints may not be equal, which makes the problem a bit more complex.

To solve this problem, we can use the concept of multivariate normal distribution. In this case, we have two variables - the projected distance on the plane and the error in the endpoint coordinates. The multivariate normal distribution takes into account the correlation between these two variables, which makes it suitable for our problem.

To find the distribution of the projected distance, we can use the formula for the marginal distribution of a multivariate normal distribution. This formula takes into account the correlation between the two variables and gives us the distribution of the projected distance.

However, this formula involves some complex mathematical calculations, and it may not be easy to understand for non-mathematicians. So, I would suggest using a statistical software like R or Matlab to calculate the distribution. These software have built-in functions to calculate the marginal distribution of a multivariate normal distribution.

I hope this helps. Good luck with your research!