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Beer-monster
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Homework Statement
I'm just checking my thinking on my understanding of probability and probability distributions as it seems a little rusty.
So let say we have a set of boxes arranged in a grid. We throw balls, all of identical mass m, onto the grid at random.First, am I right in saying that the probability of a ball landing in a given box can be given as the area of a single box a divided by the total area of the grid of boxes A.
[tex] p = \frac{a}{A} [/tex]Secondly, if I know that the mean number of balls I throw in a given time is [itex]\lambda = Np[/itex] the probability of a given box containing n balls is given by the Poisson distribution.
[tex] P(n)= \frac{\lambda^{n}}{n!}e^{-\lambda} [/tex]
If I wanted to determine the probability of the total mass inside a given box, how would I go about it.
My feeling is that as the mass is the same for all of the balls, the distribution of masses is the same as the distribution of the numbers. Therefore, can I just convert it to a function of total mass M by substitution of [itex] n=M/m [/itex] ?
i.e. [tex] P(M) = \frac{\lambda^{M/m}}{(M/m)!}e^{-\lambda} [/tex]
Is this correct? If so that factor term seems a little odd now, is there a way to neaten it.