If X and Y have a joint distribution which is uniform (constant density) on the two dimensional region R (usually R will be a triangle, rectangle, or circle in the (x,y) plane), then the pdf of the joint distribution is inside the region R (and the pdf is 0 outside). The probability of any event A (represented by a subset of R) is the proportion . Also the conditional distribution of Y given X=x has a uniform distribution on the line segment (or segments) defined by the intersection of the region R with the line X=x.
I'm not sure what this all means, but it's in bold print in my book, so I'm guessing it's important. Can someone maybe provide an example or something? Thanks