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5. In the carnival game Under-or-Over-Seven, a pair of fair dice is rolled once, and theresulting sum determines whether the
player wins or loses hisor her bet. For example, the player can bet $1 that thesum will be under 7—that is, 2, 3, 4, 5, or 6.For this bet, the player wins $1 ifthe result is under 7 and loses $1 if the outcome equals oris greater than 7. Similarly, the player can bet $1 that the sum will be over 7—that is, 8, 9, 10, 11,or 12. Here, the player wins $1 if the result is over 7 but loses $1 if the result is 7 or under. A thirdmethod of play is to bet $1 on the outcome7. For this bet, the player wins$4 if the result of the roll is 7 and loses $1 otherwise.(10%)
a. Construct the probability distribution representing thedifferent outcomes that are possiblefor a $1 bet on under 7.
b. Construct the probability distribution representing thedifferent outcomes that are possiblefor a $1 bet on over 7.
c. Construct theprobability distribution representingthe different outcomes that are possiblefor a $1 bet on 7.
d. Show that the expectedlong-run profit (or loss) to the player is the same, no matter which method of play is used.
在7以下或7以上的狂欢节游戏中，一对公平骰子掷一次，由此产生的总和决定
玩家赢或输的赌注。例如，玩家可以下注1美元，金额将低于7，即2、3、4、5或6。对于本次下注，如果结果低于7，玩家赢1美元，如果结果等于或大于7，玩家输1美元。
同样，玩家可以下注1美元，总金额将超过7，即8、9、10、11或12。在这里，如果结果超过7，玩家赢1美元，但如果结果低于7，玩家输1美元。第三种玩法是在结果7上下注1美元。对于本次下注，如果掷骰结果为7，则玩家赢4美元，否则玩家输1美元。(10%)
a、构建概率分布，代表7岁以下1美元赌注可能产生的不同结果。
b、构建概率分布，代表7美元以上1美元赌注可能产生的不同结果。
c、构建概率分布，表示1美元押注7可能产生的不同结果。
d、表明无论使用哪种游戏方法，玩家的预期长期利润（或损失）是相同的。
考点：（10分）
X P(X)
$ - 1 21>36
$ + 1 15>36
X P(X)
$ - 1 21>36
$ + 1 15>36
X P(X)
$ - 1 30>36
$ + 4 6>36
SOLUTION：
古典概率树状图法即可；随机数法
a. (d1,d2)
(1,1) (1,2)(1,3)…(1,6)
(2,1)(2,2)……….(2,6)
…
(6,1)……………..(6,6)
| SUM | Obs | Percentage |
2 | X1 | X1/36 |
3 | X2 | X2/36 |
… |
|
|
12 |
|
|
赌点数和小于7（大于等于7同理）：
EXP(OUTCOME)=1*P(SUM=2,3,4,5,6)+0*P(SUM=7,8,9,..12)=-0.167
只赌点数和为7：
EXP(OUTCOME)=4*P(SUM=7)+0*P(SUM≠7)=-0.167
b. 2
5
III. CaseStudy and Analysis（30%）
The relationship between executivecompensation and profitability of listed companies hasalways been an important research issue. The file(ceo-compensation.xlsx) lists the returns and average executive income of major U.S. public companies in 2013.
1.Please complete the do file so that the do file can estimatethe influence from the Var. Compensationto the Var. Returns： （20%）
a. Automaticallyimport the required data.
b. Keep the first Nlisted companies from the file and drop the remaininglisted companies.
(N is equal to the last two digitsof your student id)
c. Check out the influence from the Var. Compensationto the Var. Returns
d. Please explain themeaning of the regression resultsand give a conclusion.
e. Divide the program into many paragraphs and add Chinese notes to the program.
高管薪酬与上市公司盈利能力的关系一直是一个重要的研究课题。该文件（ceo compensation.xlsx）列出了2013年美国主要上市公司的回报和平均高管收入。
1.请填写do文件，以便do文件能够估计薪酬对盈利的影响：（20%）
a、 自动导入所需的数据。
b、 从文件中保留前N家上市公司，并删除其余上市公司。
（N等于学生id的最后两位数字）
c、 查看风险补偿对风险回报的影响
d、 请解释回归结果的含义并给出结论。
e、 将程序分成许多段落，并在程序中添加中文注释。
请在程序完成后将do文件上传至云班课，并在答卷纸用中文详细写出计算方法，回答上述问题，并逐行列出每一步计 算所需要公式。
2. Please check ifgender is the proper moderators（调节变量）for the relationship betweenVar.Compensation and the Var. Returns. And then:
a. Specify the methods you use, report the key statistics and give a conclusion. （5%)
b. If we want to check whether the gender was the propermediator variable（中介变量）for the relationshipbetween
Var. Compensation and the Var. Returns, what should we do?Please submit your do file, reportthe key statistics and
give a conclusion. (5%)
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