Let $n$ and $r$ be integers. Compute the number of solutions to the equation
$$x_1 + x_2 + \cdots + x_r = n,$$
where $x_i \geq 0$ are integers. Next, assuming the uniform distribution on the space of solutions, find $P(x_1 = a)$ and its limit as $r\to\infty, n\to\infty, n/r\to \rho > 0$.
The first part of the question is just stars and bars. One can easily get ${}n + r – 1 \choose r – 1$ .
Now I'm not so sure how to find $P(x_1 = a)$ or the limit.
This is from a Probability and Measure Theory book, so I am trying to prove everything formally (particularly the limit portion). I tried to come up with something to condition on, and I got nowhere. I'm pretty sure it'll be a function of $n$ and $r$ because of the limit question that follows it. Does anyone have any ideas?
Best Answer
The number of cases where $X_1=a$ is the same as the number from your original calculation but replacing $n$ by $n-a$ and replacing $r$ by $r-1$, so ${n-a + r - 2 \choose r - 2}$
and with the uniform distribution this makes $\mathbb P(X_1=a) = \dfrac{{n-a + r - 2 \choose r - 2}}{{n + r - 1 \choose r - 1}}$ which you could rewrite
I will leave finding the formal limit to you, but it would not surprise me if you ended up with a Poisson distribution with parameter $\rho$, i.e. $\lim \mathbb P(X_1=a) = \frac{\rho^a }{a!}e^{-\rho}$