Uniform Joint distribution !

[desai_hrishi]

A particular explanation in the study manual says that - "x and Y have a joint distribution which is uniform on the 2D region R. Hence, Conditional distribution of Y given X=x has a uniform distribution on the line segment(s) defined by the intersection of the region R with line X=x."

I didn't quite catch the meaning of the last line. Why does it say "uniform distribution on the 'line segment' intersecting region R" ?

[daaaave]

We are given that X=x, so we know that we are on that line. The original distribution only took on values in R, so we also must still be in R. The only points on both the line and R are the points on the line segment defined by X=x intersect R (or line segments, if R has a funky shape).

The conditional distribution is uniform because the original distribution was uniform. You can think of it as saying that each possilbe point was equally likely in the original distribution, and nothing has changed the relative weights so they are still equally likely in the conditional distribution. Or you could say that the original joint density was constant, and the new conditional density (on the set of possible values) is a constant times the old joint density and is thus still constant and hence uniform.

[desai_hrishi]

Thanks for putting that in lucid language. I got it now!

[mathmajor]

Very nice explanation. Quick aside: That's the best way to teach math. Plain-language explanations with a step-by-step thought process.

[Kaner3339]

Watch Dave's videos and there's a good example or two of this. Really intuitive when you see it graphically.