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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 ?)

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I believe this problem is discussed in Tuckwell's "Elementary Applications of Probability Theory", section 3.6. –  Sasha May 9 '12 at 20:44
    
You are probably expected to exploit the relationship between the Poisson distribution and the exponential distribution. It is usually expressed in terms of time. If the number of events in unit time is Poisson mean $\lambda$, then interarrival times are exponential mean $1/\lambda$. In our case, volume plays the role of time. So you can find an expression for the probability that asphere of certain volume about $x$ is empty, and turn this into information about distance. –  André Nicolas May 9 '12 at 20:47

2 Answers 2

up vote 1 down vote accepted

1) Yes. $\ \ $ 2) $\mathbb R_+$.

A possible choice of probability space $(\Omega,\mathcal F)$ is as follows:

  • each $\omega$ in $\Omega$ is a locally finite subset of $\mathbb R^3$,
  • $\mathcal F$ is the smallest sigma-algebra such that counting functions of Borel subsets are measurable.

In other words, one asks that $[N_B=k]$ is in $\mathcal F$, for every Borel subset $B$ of $\mathbb R^3$ and every integer $k$, where $N_B(\omega)=\#(\omega\cap B)$.

Then $X:(\Omega,\mathcal F)\to(\mathbb R_+,\mathcal B(\mathbb R_+))$ is a random variable since $[X\gt r]=[N_{B(r)}=0]$ where, for every real number $r\gt0$, $B(r)$ is the ball in $\mathbb R^3$ with radius $r$ and centre $x$.

And $\mathrm P(N_{B(r)}=0)$ is not difficult to compute since one knows the distribution of $N_{B(r)}$.

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1) Yes, it's well defined. You should see it if your are familiar with analogous Poisson process in the real line. In that case, you have a point process with "intensity" $\alpha$, so that if you take a very small interval $dx$ the probability that an event occurs in that interval is $p = \alpha \; dx$, and the events ocurrences in different intervals are independent. Hence, letting $dx \to 0$ you get that the number of events in any interval of length $L$ (actually, in any measurable subset of $\mathbb{R}$) is a Poisson variable with expected value $\lambda = \alpha L$. The same happens here, in $\mathbb{R}^3$.

2) You must pick a point at random in $\mathbb{R}^3$ and measure the distance to the nearest event ("star"). $X$ takes values in $(0,+\infty)$, and it's not difficult to compute, considering that $P(X \ge x)$ is just the probability that a ball of radius $x$ has zero events.

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