Given X~Poi(λ) and that X balls are dropped into n bins, what does "X balls" exactly refer to?

Question

Problem:

Let $X \sim \operatorname{Poi}(λ)$. Given $X$, suppose that we drop $X$ balls uniformly at random into $n$ boxes. What is the unconditional distribution of the number of balls in box $1$?

 

My Question: I was not sure what the problem meant by "$X$ balls" being dropped since 1) $X$ is defined as a random variable with a Poisson distribution and 2) the balls are defined as being dropped "uniformly at random" (i.e. with $1/n$ probability). 

 

Does this mean that a) the probability of each ball falling into the boxes has the Poisson distribution or b) the total number of balls falling is determined by the Poisson?

Answer 1

The keyword here is given. When we are told that something happens given $X$, that means that we are considering the conditional distribution given the event that $X=x$ for som $x \in \mathbb{N}_0$. And given that we know that $X=x$, it is most certainly possible to distribute $x$ balls uniformly in $n$ bins.

If you know that the total number of balls is $x$, then the number of balls in the first bin, which we will refer to with a random variable $Y$, would follow a $\operatorname{binomial}(x,\frac{1}{n})$ distribution. Written in symbols the setup can be formulated as $$X \sim \operatorname{poisson}(\lambda)$$ $$Y | X=x \sim \operatorname{binomial}(x,\frac{1}{n})$$ The task is then to find the unconditional distribution of $Y$, which can be done with the law of total probability: $$\mathbb{P}(Y=y) = \sum_{x=0}^\infty \mathbb{P}(Y=y \: | \: X=x)\mathbb{P}(X=x).$$

Answer 2

This is a two-step procedure.

• First you select the number of balls using a Poisson distribution $Poi(\lambda)$. This random number of balls is denoted by $X$.
• Then, once you know you have $X$ balls (now you consider this number as fixed), you assign each ball to one of the $n$ boxes, uniformly at random.

So proposition b) is correct.