Approaching the Confusing Ball Problem: Independent Distribution and Joint PDF

[sezmin]

hey, I am having a lot of trouble approaching this problem:

let m balls be independently distributed into s boxes with equal probabilities
m has a poisson distribution with mean 1/b
let Xi be the number of balls in box i, i = 1,2,...,s
let y = the number of empty boxes
are X1, X2, ... , Xs independent?

i can't really conceptualize a function to describe the Xi's and can't find the joint pdf of X1,X2,...Xs

my thought was that p(Xi) = (1/s)^(Xi) * ((s-1)/s)^(m - Xi)
but, i don't see how that gets anywhere...i would appreciate any advice!

 

[lippyka]

The short answer is that the Xi's are not independent. To understand why, we need to look at the joint probability distribution of X1, X2, ... , Xs. In this case, the joint probability distribution of the Xi's follows a multivariate Poisson distribution with mean vector (1/b, 1/b, ..., 1/b). This means that the joint probability of observing a particular set of values for the Xi's can be calculated using the formula for the multivariate Poisson distribution. What this also means is that the Xi's are dependent on each other, since their joint probability distribution is not uniform. For example, if X1 is larger than expected, then X2 and X3 are likely to be smaller than expected since the mean vector has a fixed magnitude (1/b). Finally, it is worth noting that the number of empty boxes (y) is also dependent on the values of the Xi's. Since y = m-Sum(Xi), the value of y will be affected by the values of the Xi's. Therefore, we can conclude that the Xi's are not independent.