Question: 2015 Fall Final - 19

Author: Michiel Smid

The $n$ students $S_1,S_2,\dots,S_n$ decide to organize a Secret Santa: They take a uniformly random permutation $P_1,P_2,\dots,P_n$ of $S_1,S_2,\dots,S_n$. For each $i = 1,2,\dots,n$, student $S_i$ buys a gift and gives it, anonymously, to student $P_i$.

For each $i = 1,2,\dots,n$, let $v_i$ be the value (in dollars) of the gift that student $S_i$ buys. Let $Y$ be the value of the gift that student $S_1$ receives, and let $Z$ be the value of the gift that student $S_2$ receives. What is $\mathbb{E}(2 \cdot Y - Z)$?

(a)

$2 \sum_{i=2}^{n} v_i/n - (v_1/n + \sum_{i=3}^{n} v_i/n)$

(b)

$2v_1 - v_2$

(c)

$\sum_{i=1}^{n} v_i/n$

(d)

$\sum_{i=3}^{n} v_i/n$