0
$\begingroup$

I have to model the given situation: Let us say there is an sphere given by volume $V_r$, which is space basically. It is continuously bombarded with balls, now it is assumed that the process of coming a ball inside a sphere is a volumetric Poisson counting process. Suppose the sphere is moving with Gaussian distribution, now what is the whole process of coming of balls becomes:

  1. I think that it should be the $z=yx$ problem where y is Poisson distribution and x is Gaussian distribution and I need to find the density of z, which I can do. Am I rightly interpreting this question?

  2. How to calculate the Bayesian estimator of z if we have the given series of measurements of coming of balls inside the space sphere, given that the measurements are independent and discrete in time.(I mean to say can it have a Conjugate prior)

Please also mention the references and from where I can learn such things.

  • 0
    What do you mean "moving with Gaussian distribution?" That its speed in $\mathbb{R}^3$ is Gaussian? – 2017-02-21
  • 0
    The position of the center of the sphere is Gaussian distributed, or to be more precise the sphere is subjected to Brownian motion. – 2017-02-22

1 Answers 1

0

Think about how many balls you collect in a time $\Delta t=t_1-t_0$. Instead imagine a very thin cylinder, essentially in $\mathbb{R}^1$, that only collects balls on either end. If in a time interval $\Delta t$ you move at velocity $c_1>0$ for $t$ on the time interval $[t_0,t_1]$, and the balls are coming from $-\infty$ and $\infty$ at a speed $\gamma$, then we collect any ball on the right if it was $\leq (\gamma+c_1)\Delta t$ away from the cylinder, and any ball from the left if it was $\leq(\gamma-c_1)\Delta t$ away from the cylinder. Assuming the balls are infinitely small and everywhere, we collect $2\gamma\Delta t$ balls if $c_1<\gamma$. This translates to being bombarded at a rate of $2\gamma$, regardless of $c_1$ as long as $c_1<\gamma$. Usually we think of these balls as neutrinos or waves of light so large $\gamma$ is acceptable. We can extend this to any object whose velocity changes only finitely many times, and then to any object whose velocity is bounded and measurable. So in essence speed doesn't matter and we can extend this reasoning to the sphere, in fact any $n$-sphere in $\mathbb{R}^{n+1}$.

The problem with Brownian motion is that it doesn't have a bounded speed, $\limsup_{t\rightarrow\infty}\frac{B_t}{t}=\infty$ with probability 1, so we can't repeat our old work for finite $\gamma$. For the sphere, what this might mean is that infinitesimally the sphere can travel so fast over a small interval that it leaves a vacuum behind it, so that overall it's getting hit less. But it would be a nightmare to try to analyse that.