Average distance between consecutive uniformly distributed values

Question

Say I have a list of values $(X_1, X_2, ..., X_n)$ and $X_i$ is a uniform random variable between 0 and 1.

Let $(Y_1,Y_2,...,Y_n)$ be the ordered list of those values. How do I find the expected distance between $Y_i$ and $Y_{i+1}$?

Answer 1

You are seeking to find $$\mathbb{E}\left(\vert Y_{i+1} - Y_{i} \vert \right) = \mathbb{E}\left( Y_{i+1} - Y_{i} \right) = \mathbb{E}\left(Y_{i+1}\right) - \mathbb{E}\left(Y_i\right) = \frac{i+1}{n+1} - \frac{i}{n+1} = \frac{1}{n+1}$$

The expectation of $Y_i$ is easy to find, noting that $Y_i \sim \mathrm{Beta}\left(i, n+1-i\right)$: $$ \mathbb{E}\left(Y_i\right) = \frac{a}{a+b} = \frac{i}{n+1} $$