Given numbers $(1,2,3,4)$ what is the expectation of the sum of the first and last number? I know that for both the first and the last number the expectation for each number will be 10/4, however short of writing down all the permutations and counting the sums and then getting the expectations that way, how would I express $E(X)$ given that $X$ is the sum of first and last digits of the permutation?
Thanks!
Let $Y$ be the first digit of the permutation and $Z$ be the last digit of the permutation. Then $X=Y+Z$.
$Y$ takes on the values $1,2,3,4$ with equal probability. (Why?) Similarly $Z$ takes on the values $1,2,3,4$ with equal probability.
However, $Y$ and $Z$ are clearly not independent (if $Y=2$ then we know $Z \ne 2$).
Nevertheless, a useful property of the expectation is $$E[Y+Z] = E[Y] + E[Z]$$ even if $Y$ and $Z$ are not independent.
Thus, since $E[Y]=E[Z]=\frac{1+2+3+4}{4}=2.5$, we have $E[X] = 2.5 + 2.5 = 5$.