Distribution of ordered random variables

Question

Let $\Phi$ be a homogeneous PPP of intensity $\lambda$ on $\mathbb{R}$. Let {$x_0, x_1, \dotsc$} $= \Phi~\cap~\mathbb{R}^+$, in increasing order, i.e., $0 < x_0 < x_1 < \dotsc$. I want to find the complementary cumulative distribution function (CCDF) of $x_0/x_k$, where $x_k$ is Erlang distributed with pdf $f_{x_k}(x) = \frac{\lambda^{k+1}x^k}{k!}e^{-\lambda x}$.

Answer 1

Got the answer! Since we have a homogeneous PPP, the random variables $x_0/x_i$ with $i = 1, 2, \dotsc$ are uniformly distributed in the interval (0, 1). Also, since $x_1 < x_2 < \dotsc < x_k$, we have $x_0/x_1 > x_0/x_2 > \dotsc > x_0/x_k$. Thus, $$\mathbb{P}\left(\frac{x_0}{x_k} > \theta \right) = \mathbb{P}\left(\frac{x_0}{x_1} > \theta, \frac{x_0}{x_2} > \theta, \dotsc, \frac{x_0}{x_{k-1}} > \theta\right).$$

Since $x_0/x_i$ are independent random variables, we have $$\mathbb{P}\left(\frac{x_0}{x_k} > \theta \right) = (1-\theta)^{k-1}.$$