If I throw $n$ points independently at random in the real interval $[0,1]$, what is the probability distribution for the size of the maximum gap between any two consecutive points?
I am assuming the points are sorted when looking for consecutive points.
The joint pdf of the ordered statistics $X_{(1)}, X_{(2)}, \ldots, X_{(n)}$ is
$$ f_{X_{(1)}, X_{(2)}, \ldots, X_{(n)}}(x_1, x_2, \ldots, x_n) = n!, 0 < x_1 < x_2 < \ldots < x_n < 1$$
We are interested in the spacing of the ordered statistics $Y_i$, defined by $$Y_i = X_{(i)} - X_{(i-1)}, i = 2, 3, \ldots, n $$
Together with the auxiliary transform $Y_1 = X_{(1)}$, we can compute the joint pdf of $(Y_1, Y_2, \ldots, Y_n)$. Note that the inverse transform is given by $$X_{(i)} = \sum_{j=1}^i Y_{j}, i = 1, 2, \ldots, n$$ and thus the Jacobian is equal to $1$. Therefore the joint pdf of $(Y_1, Y_2, \ldots, Y_n)$ is
$$ f_{Y_1, Y_2, \ldots, Y_n}(y_1, y_2, \ldots, y_n) = n!, 0 < y_i < 1, \sum_{i=1}^n y_i < 1 $$
Since the support is a simplex in $\mathbb{R}^n$, so we can see the normalizing constant is alright. If you consider the spacing at the two end, i.e. $X_{(1)} - 0 = Y_1$ and $1 - X_{(n)} = 1 - \sum_{i=1}^n Y_i $, then we define the maximum to be
$$ Z = \max\left\{Y_1, Y_2, \ldots, Y_n, 1 - \sum_{i=1}^n Y_i\right\}$$
Since the last term is linearly dependent on the previous term, so we actually has a $n$ dimensional randomness only. Consider the CDF
$$ \begin{align*} F_Z(z) & = \Pr\{Z \leq z\} \\ & = \Pr\left\{Y_1 \leq z, Y_2 \leq z, \ldots, Y_n \leq z, 1 - \sum_{i=1}^n Y_i \leq z \right\}\end{align*}$$
with $\displaystyle z \in \left[\frac {1} {n+1}, 1\right]$. It involve the integration with a smaller simplex. Sorry I do not have enough time to try this at the moment. Maybe someone can finish this off.
I am assuming the points are sorted"... changes the question entirely. – wolfies Dec 24 '15 at 16:47