TITLE: Limiting case of Binomial(n,p)/n?
QUESTION [5 upvotes]: Let the random variable $X$ have distribution $X \sim \text{Binomial}(n,p)$.  Let $Y = X/n$.  What is the limiting distribution of $Y$, as $n \to \infty$?  Does it have a simple distribution?
Of course, when $n$ is large, $X$ has approximately the distribution $\text{Poisson}(np)$.  Thus, we could ask the question in the following alternative way: suppose $X^* \sim \text{Poisson}(np)$, and define $Y^* = X^*/n$; what is the limiting distribution of $Y^*$, as $n \to \infty$?
I have not been able to find an existing result on this, though it sounds like the sort of thing that someone must have studied long ago.  When I search for limiting distribution and Poisson or limiting distribution and Binomial, I find many references to the fact that $\text{Binomial}(n,p) \to \text{Poisson}(np)$ as $n \to \infty$, which I already knew, so I'm not sure where to look to figure this out.

REPLY [2 votes]: A standard approximation for the binomial distribution says that $X$ has approximately a Gaussian distribution, i.e., $X \sim \mathcal{N}(pn,p(1-p)n)$.  It follows that $Y$ also has approximately a Gaussian distribution, i.e., $Y \sim \mathcal{N}(p,p(1-p)/n)$.  This approximation becomes better and better as $n \to \infty$.  The limit as $n \to \infty$ is a Dirac delta at $p$.