ผลต่างระหว่างรุ่นของ "Probstat/midterm problem 9 solution"
Jittat (คุย | มีส่วนร่วม) |
Jittat (คุย | มีส่วนร่วม) |
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แถว 33: | แถว 33: | ||
Note that this slightly larger than 1/4 of the people. | Note that this slightly larger than 1/4 of the people. | ||
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+ | '''Remarks''' In the solution above, we analyze the experiment in terms of the people, so there are <math>2n</math> people that we need to look at. Another way is to analyze the experiment in terms of pairs. In that case, there will be <math>n</math> pairs. The second approach works fine and the answer will be the same. | ||
== Variance == | == Variance == |
รุ่นแก้ไขเมื่อ 08:33, 1 ธันวาคม 2557
This is how a simple charity game is played. There are people where . There are also tickets whose numbers on each of them is either 0 or 1. There are n tickets with number 1 on them, and tickets with number 0 on them. The ticket are randomly given to the people; each person gets one ticket. (So that people get tickets number 1 and the other 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.
Expectation
(a) Let random variable denote the amount of money donated. Find . Show your work clearly.
Consider person . Let random variable
Note that . Person donates if and only if the person gets ticket number 0, and the person's pair get ticket number 1. This occurs with probability
Since
using the linearity of expectation we have that
Note that this slightly larger than 1/4 of the people.
Remarks In the solution above, we analyze the experiment in terms of the people, so there are people that we need to look at. Another way is to analyze the experiment in terms of pairs. In that case, there will be pairs. The second approach works fine and the answer will be the same.