# Distribution of gaps between uniform random variables

Pick $$k$$ uniform independent random integers $$x_1,\dots,x_k\in\{0,1,\dots,2^t-2,2^t-1\}$$ and denote $$y_{\sigma(i)}=x_i$$ where $$\sigma$$ is a permutation in $$S_n$$ such that $$y_1\leq y_2\leq\dots\leq y_{k-1}\leq y_k$$ holds.

Define the gaps $$g_1,\dots,g_{k-1}$$ by $$g_i=y_{i+1}-y_i$$.

1. What is the probability distribution $$P(\sum_{i=1}^{k-1}g_i^2)$$?

2. What is the distribution that $$g_{max}-g_{min} holds at an $$r\geq0$$?