Suppose, on $\mathbb{R}^2$, that $X$ is a random variable which takes values uniformly at random over the $\textit{line segment}$ from $(0,0)$ to $(a,a)$, where $a > 0$ is a positive constant.
How can one find the distribution of $X$ over $\mathbb{R}^2$?
It seems that if we try to "project" $X$ to one-dimension, then we show get the uniform distribution over $\mathbb{R}$. But then, for example, we cannot conclude that \begin{equation} \mathbb{P}[X \in \text{line segment from $(0,0)$ to $(b,b)$}] = b/\sqrt{2}? \end{equation}
Anyone gets an idea? Thanks very much.