# Haunted by Cauchy – A medium level problem

#### Forest Finder

##### New Member
Hi

Let's suppose X1 and X2 are two Normally distributed variables. [In general, X1 is from N(m1,s1^2) and X2 from N(m2, s2^2) where m1<m2 and s1^2≠s2^2.] However, let’s say, for the sake of starting the discussion, that both are Normally distributed N(1,2^2) and N(2,3^2). One of X1 or X2 is random. But we have an interesting constraint, being that the variance of X1/X2 samples ‘is equal to’ a predetermined finite constant c (i.e. we exclude an adjustable ‘small n’ interval containing x2=0; e.g. excluding no more than say 0.01% of the sample at the extreme tails of the x1/x2 Cauchy distribution); say, c=2

Now the question is, what relationships are there between X1 and X2 that are implied by this constraint (e.g. are they correlated, or would they be correlated if other conditions also apply?), and how do these relationships (if there are any) change with the size of the constant c?

Personally, when I can’t do the math to work out these things, I usually do a simulation to give me some ideas. I tried a simulation with the following steps (starting by choosing x1) but got stuck on step 3 'choosing x2 that meets the variance condition': Is it even possible to do this?
1. Select a random x1 from, say, N(1, 2^2)
2. Preselect the variance of our X1/X2 sample to be, say, 1.5
3. How can we select x2 from, say, N(2,3^2) such that our x1/x2 samples tend to have a variance of 2?
What then are your thoughts on possible (x1, x2) relationships or the specter of impossibility?