A uniform mixture of order statistics

Question

Let $0<k<n$ be integers, and let $X$ be a random variable obtained as follows: sample $n$ points independently and uniformly at random in the unit interval, and select (uniformly) one of the $k$ leftmost points. The distribution of $X$ is therefore a uniform mixture of order statistics, with pdf given by $$f(x) =\frac{1}{k}\sum_{i=1}^{k}n\binom{n-1}{i-1}x^{i-1}(1-x)^{n-i} $$ At the bottom, I draw a picture of the pdf for $n=20$ and $k=9$.

My question is, are there any simpler expressions that approximate this well? It has a natural S shape to it so I wonder if it is similar to a logistic curve, for example.

Answer 1

For $x\in(0,1)$, $$f(x)=\frac nk\,F_{n-k,k}(1-x),$$ where $F_{n-k,k}$ is the cdf of the beta distribution with parameters $n-k,k$. From here, you can get a number of approximations. See e.g. Wikipedia.

Also, you can use the central limit theorem for the beta distribution (based on the delta method or the asymptotics of the beta integral; cf. MathSE) to get the following: If $n\to\infty$, $k\sim an$ for some $a\in(0,1)$, and $(a-x)\sqrt n\to y\in\mathbb R$, then $$f(x)\to\frac1a\,\Phi\Big(\frac y{\sqrt{(1-a)a}}\Big),$$ where $\Phi$ is the standard normal pdf.

Here are the graphs $\{(y,f(a-y\sqrt n))\colon|y|<3\sqrt{(1-a)a}\}$ (red) and $\{(y,\frac1a\,\Phi\big(\frac y{\sqrt{(1-a)a}}\big))\colon|y|<3\sqrt{(1-a)a}\}$ (blue) for $n=20$, $k=6$, and $a=k/n=.3$: