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I am studying a modelling problem which is given as: Suppose I have a sphere(empty) and there are some particles that are entering it with Poisson distribution, and it is given that the number of incoming particles is given by: $y_i$~$Poiss(\rho)$. $\rho$ is given as rate of occurrence. Now if the center of sphere is also distributed or moving with Gaussian distribution, then what will be the resulting distribution of number of incoming particles?

Also it is given that for this condition the Bayesian estimator of $\rho$ is Gamma distribution(when only Poisson distribution is there), now what is Bayesian estimator of the distribution(where both normal and Poisson are considered) in question?

Please also tell me the resource where I can get to know such modelling.

PS: I have asked somewhat same question on mathematics stack Exchange site but no answer was provided, and being a member with low reputation, I can't transfer question.

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  • $\begingroup$ Not sure tu fully understand your question, but the rho must depend on the position of (the center of) the sphere I assume? Do you know mixtures of distributions? $\endgroup$
    – user83346
    Commented Feb 22, 2017 at 7:09
  • $\begingroup$ You get a question migrated by flagging and asking. But I expect your question wasn't answered there because the situation is unclear; does moving the center of the sphere affect the distribution of incoming particles? Why? In what way? $\endgroup$
    – Glen_b
    Commented Feb 22, 2017 at 7:09
  • $\begingroup$ @ fcop I do not know about the mixture of distribution, can you please give me a reference book or paper for the same $\endgroup$
    – Userhanu
    Commented Feb 22, 2017 at 7:23
  • $\begingroup$ @ Glen_b "does moving the center of the sphere affect the distribution of incoming particles? Why? In what way?" How should I determine that? I am sorry but i am completely lost? $\endgroup$
    – Userhanu
    Commented Feb 22, 2017 at 7:28
  • $\begingroup$ We are assuming that sphere is surrounded by constant density of particles, should the distribution get affected then ? $\endgroup$
    – Userhanu
    Commented Feb 22, 2017 at 7:41

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Looks like a homework problem.

When the incoming particles $y_i$ ~ Poisson $(\rho)$, $\forall i \in [1, n]$, then a $\Gamma\ (\alpha, \beta)$ prior on $\rho$ is a conjugate prior, and the posterior distribution is then a $\Gamma \ (\Sigma y_i + \alpha, n+\beta)$. To understand this, look up the Poisson-Gamma mixture model, its connection to the Negative Binomial distribution. Then, look up the Normal approximation to the Poisson distribution. On a different note, do you see the relationship between Normal and Gamma distribution here?

Particle counting is the literature that often presents such problems. Alternatively, they come up in Actuarial statistics, queueing theory, and several other domains. Here is one such example.

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  • $\begingroup$ Just a small help, do you know any book or resources apart from the one you have mentioned that can help me strengthen my base in this area(some probability book or something like that). Or should I start with books on Actuarial statistics, queueing theory!! $\endgroup$
    – Userhanu
    Commented Feb 22, 2017 at 9:09
  • $\begingroup$ Any book on probability theory should be good. If it's your first course, then, you may consider (a) Mathematical Statistics with Applications by Wackerly, Mendenhall and Scheaffer, and (b) A First Course in Probability along with Introduction to Probability Models by Sheldon Ross. $\endgroup$
    – dangiankit
    Commented Feb 22, 2017 at 9:17
  • $\begingroup$ I have one doubt you said to look for "Normal approximation to the Poisson distribution" but what I am asking that if the location of sphere center is Gaussian distributed, (moving with Brownian motion), should this not accounts to having a Gaussian distributed error in the Poisson Process. $\endgroup$
    – Userhanu
    Commented Feb 24, 2017 at 11:28
  • $\begingroup$ Because "Normal approximation to the Poisson distribution" is to study the Gaussian probability distribution as a special case of the Poisson distribution applied to measurements with a large mean value-quoted. Now will the mean(rate of arrival of particles) becomes larger when we have sphere position as Gaussian distributed? $\endgroup$
    – Userhanu
    Commented Feb 24, 2017 at 11:30

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