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Consider a Poisson process with rate 1 over $R_+ \times R$. I found this term in a scientific article, I assume it means :

-The number of points in a subset $A$ of $R_+ \times R$ has the Poisson distribution with paramater the area of $A$, and this random variable (number of points in $A$) is independant of the number of points in $B$ if $B$ is disjoint set from $A$.

Is this the right definition ?

My main question : it seems that with probability 1, no two points can have the same x-coordinate. Therefore, we can order the points by their x-coordinate : $(X_1,Y_1),(X_2,Y_2)$ etc what is the distribution of say the y-coordinate of the first point (the point with smallest x-coordinate)?

I would also want to check if my assumptions are right : the y-coordinate are independant variables, and the x-coordinate form a Poisson Process over $R$.

Note : I have heard about the ****2d**** Poisson process over $R_+ \times [0,M]$ where $M$ is some number, which can be constructed as a marked Poisson process, so the answer to my questions would be yes, but the problem is that here I have $R_+ \times R$ instead of $R_+ \times [0,M]$ so I can't see my process as a marked Poisson process

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    $\begingroup$ Why should there even exist a point with smallest $x$ coordinate? I believe there is zero chance of this happening, because when it does happen and that coordinate is $x_0\gt 0,$ for all $A\ge 0$ every rectangle $(0, x_0)\times (-A,A)$ has no points and the chance of that is $\exp(-2Ax_0)\to 0$ as $A\to\infty.$ $\endgroup$ – whuber Apr 9 at 19:22

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