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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.

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  • $\begingroup$ The question is not "how" but "where". You can find the distribution in your first sentence. $\endgroup$ – zhoraster Nov 1 '15 at 17:58
  • $\begingroup$ What do you mean by "distrubution over $\mathbb R^2$"? This distribution is not absolutely continuous, so there is no probability density function. $\endgroup$ – GEdgar Nov 1 '15 at 18:04
  • $\begingroup$ @GEdgar For example, I want to compute that probability $\mathbb{P}[X \in \text{line segment from $(0,0)$ to $(b,b)$}]$, for some $0 < b < 1$. $\endgroup$ – Richie Nov 1 '15 at 18:09
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    $\begingroup$ @GEdgar even it has no probability density, any random variable on $\mathbb R^n$ has a distribution... $\endgroup$ – user251257 Nov 1 '15 at 18:14
  • $\begingroup$ What is the distribution of $X=U\cdot (a,a)$ where $U$ is uniformly distributed on $[0,1]$? $\endgroup$ – user251257 Nov 1 '15 at 18:17

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