Question: 2016 Fall Final - 19

Let $n \geq 2$ be an integer and let $a_1a_2 \dots a_n$ be a uniformly random permutation of the set $\{1,2,\dots,n\}$. Let $X$ be the random variable with the value

• X = the number of indices $i$ with $1 \leq i \leq n - 1$ and $a_i < a_{i + 1}$.

For example, if $n = 6$ and the permutation is 3, 5, 4, 1, 6, 2, then $X = 2$.
What is the expected value $\mathbb{E}(X)$ of $X$?
Hint: Use indicator random variables.