Probstat/midterm problem 9 solution

This is how a simple charity game is played. There are $2n$ people where $n\geq 1$. There are also $2n$ tickets whose numbers on each of them is either 0 or 1. There are n tickets with number 1 on them, and $n$ tickets with number 0 on them. The ticket are randomly given to the people; each person gets one ticket. (So that $n$ people get tickets number 1 and the other $n$ people get ticket number 0.)

Then the people are randomly paired up into n pairs. In each pair the person who has lower number should donate the difference to the charity fund. The person with higher number does not donate. For example, if two people in a pair have ticket numbers 1 and 0, the person with ticket number 0 has to donate 1 baht. If two people in a pair have the same number, they do not donate.

Note: In this question, do not have to try too hard to reduce your answers to the minimal form. Any reasonable expressions are acceptable.

(a) Let random variable $X$ denote the amount of money donated. Find $E[X]$. Show your work clearly.

Consider person $i$. Let random variable

$X_{i}=\left\{{\begin{array}{ll}1&\mathrm {if} \ \mathrm {person} \ i\ \mathrm {donates} \\0&otherwise\end{array}}\right.$