Explanation for "jointly pdf is constant but marginal pdf is not"

Question

Consider: $X,Y \sim \text{uniformly distributed in }(0 \leq y \leq x \leq 1)$

From short computation, we know:

1. Jointly pdf: $f_{XY}(x,y) = 2$
2. Marginal pdf of $x$: $f_{X}(x) =\int_0^x 2\,dy=2x, 0\leq x \leq 1$

We know this is a uniformly distributed random variable, so we expect the pdf is constant.
However, the second one is not. If it is not uniformly distributed, then what type of distribution of it?

How to explain it in a more intuitive but persuasive way without using computation result?

Answer 1

The pair $(X,Y)$ is uniformly distributed in a triangle. That does not imply that either $X$ or $Y$ is uniformly distributed. Draw the triangle, and you'll see why $X$ is more likely to be in a short interval near $1$ than in an interval of the same length near $0$, so $X$ will not be uniformly distributed. Similarly, $Y$ is more likely to be in a short interval near $0$ and in an equally long interval near $1$.