Distribution of the Ratio of i.i.d Uniform Random Variables Conditional on the Sum

Suppose X,Y are both Uniform on [0,1] and indpendent. I am interested in the distribution of X/Y conditional on X+Y=constant. (Obviously, the constant must be <2.)

My hunch is that that distribution is essentially the same as the unconditional distribution of X/Y, but I have a hard time proving it.

I don't think they are identically distributed, so it's a good thing you're having trouble proving your conjecture. :)

Consider the full, x-y domain, the unit square in the x-y plane with each point equally likely (uniform density). Your conditional then splits this domain along the diagonal from (0,1) to (1,0).

When you condition your distribution of X/Y on the lower triangle (x+y<1), you will generate large values of X/Y all along (near) the x-axis (when y is small, x/y blows up).

When you look at the upper triangle that you see you have a much lower density of these points that generate large values: just in the corner near (1,0) as opposed to all along the x-axis.

So, to summarize, your conditional (x+y<1) probability will have a distribution more heavily weighted to higher values than the overall distribution (which includes the upper triangular domain where the density is weighted to lower values of x/y).