I want to solve the following exercise:
Suppose the distribution of the star in space is a Poisson-distribution, i.e. the probability that there are $n$ stars in a region $T\in\mathbb{R}^{3}$ is $e^{-\lambda}\frac{\lambda^{n}}{n!}$, where $\lambda$ is proportional to the volume of $T$. We choose randomly a point $x\in\mathbb{R}^{3}$. Let $X$ be the distance of $x$ to the next star. Then calculate the probability distribution function of $X$, i.e. calculate $P\left(s\leq X\leq t\right)$.
My questions are:
1) Is this problem even well-defined ?
Saying that stars in some region $T\in\mathbb{R}^{3}$ are Poisson-distributed (shortly: "P-distr.") confuses me, since $T$ is not fixed, so actually I have for every possible volume a different P-distr.. Thus, if the next star is for example within distance $A$ of $x$ I can have many differently shaped $T$'s with different volumes that contain $x$ at the " center" (whatever that is) and that star, so I have different P-distributions that measure my distance (Intuitively suppose I should take balls around $x$, but this explanation is not rigorous).
Conclusion: Not having a "fixed unit" $T$ with which to measure distance, makes this problem not well-defined?
2) What is the image of $X$ ? Heck, what is even our probability space ?
Possible (but very unsure) explanation: Since we dealt only with discrete (countable) probability models so far, I assume that we somehow have to approximate $\mathbb{R}^{3}$ by $\mathbb{Q}^{3}$ or $\mathbb{Z}^{3}$ (which are still countable...) and take that as our probability model $\Omega$ (I think that by symmetry we could assume $x$ to be the origin) and take our $X:\mathbb{Q}^{3}\rightarrow\mathbb{R}$ as mapping $\left(x,y,z\right)\mapsto\sqrt{x^{2}+y^{2}+z^{2}}$. This explanation would at least coincide with the fact that this exercise asks only for $P\left(s\leq X\leq t\right)$ instead of precisely $P\left(X=q\right)$ (although the wording " distribution" would rather mean the latter, I think...), since if have approximated the exact location of the star by a rational number, to make the theory work. But this seems also sketchy to me, since I don't know how to approximate the error (what should $s,t$ be ?)
(Or, a different line of thought; we accept only rational numbers as distances; but what would $\Omega$ the be ?)