这个作业是计算骰子概率和降雨概率问题

QBUS5002: Quantitative Methods for Accounting
Assignment (Part 1)

1.滚动三个模具。考虑一个游戏,玩家掷出三个标准的6面骰子。最终成绩
玩家收到的是三个骰子中两个最高数字的总和。例如,如果播放器
掷出三个骰子并得到1、3和5的分数,最终分数将为8。
每个骰子都相同,则最终得分为零。例如,如果玩家滚动
三个人死亡,得分分别为6、6和6,最终得分为0。
(a)(2分)提供了一张球员最终得分的概率分布图
游戏。
(b)(2分)计算球员最终得分分布的均值和方差
(a)部分。
(c)(3分)假设游戏相同且独立地玩了两次,找到并
绘制两个游戏的平均最终得分分布。比较分布
两场比赛的平均最终得分,其中一部分发现了一场比赛的分布
(一个)。
(d)(3分)在(c)部分中求出最终均值的均值和方差
将两个游戏的得分进行比较,并将其与(b)部分中的均值和方差进行比较。
(e)(4分)如果原始游戏重复播放100次,并且平均最终得分超过
计算所有100场比赛,并有理由讨论该平均值的分布可能如何
被近似,并且需要任何假设/条件。绘制此近似分布
并讨论如何将其与(a)和(c)部分中的分布进行比较。
(f)(2分)假设您要玩100场比赛,然后取最终分数的平均值
从所有100场比赛中删除,然后重复多次。使用近似分布
在e)中估计一个间隔,您可以在该间隔内期望最终分数的平均值的90%
说谎的100场比赛。
2. Event Management. An events manager is planning to hold a festival in Sydney for the entire month
of July in 2022 and asks the weather expert for some advice on the likely number of days it could
rain which could impact the number of people attending the festival. The weather expert notes
that historically 21% of all days are rainy days for the month of July in Sydney. The weather
expert believes that the number of rainy days in July in 2022 could be modelled with a Binomial
distribution.
(a) (2 points) Discuss the requirements of the Binomial distribution and whether you feel it is
appropriate in the current scenario.
(b) (3 points) Assuming the Binomial distribution is appropriate for modelling the number of rainy
days in July 2022, determine the probabilities of the following:
i Exactly 8 rainy days in July next year
ii Less than 9 rainy days of rain in July next year
(c) (2 points) What is the expected number of rainy days in Sydney during July next year and
what is the variance of the number of rainy days?
The events manager is also concerned about the possibility of serious transport disruptions to
the Sydney rail network, which can seriously impact the number of people able to attend the
festival. The events manager contacts Transport for NSW who inform her that serious disruptions typically occur at a rate of 1 per 40 weekdays. On weekends, disruptions are less frequent
occurring at a rate of 1 per 80 days.
(d) (2 points) The events manager believes the number of disruptions on weekdays and weekends
can be separately modelled as Poisson random variables. Discuss whether you feel the choice
of the Poisson distribution is appropriate for modelling the number of disruptions on either
weekdays or weekends.
(e) (3 points) Assuming the Poisson distribution is appropriate for modelling these disruptions,
determine the probabilities of the following:
i No more than 4 transport disruptions on weekdays during July next year
ii Zero transport disruptions on weekends during July next year
(f) (2 points) Find the mean and variance for the number of weekday disruptions in July next year.
(g) (2 points) Find the mean and variance for the number of weekend disruptions in July next year.
(h) (4 points) The events manager believes the level of profit from the festival is linked to both the
amount of rain and the number of transport disruptions in the following way:
P = 500 − 24R − 36D − 40E
where P is the amount of profit measured in thousands of dollars, R is the number of rainy days,
D is the number of weekday transport disruptions, and E is the number of weekend transport
disruptions all in July next year. Calculate the expected profit for the festival, as well as the
standard deviation of profit, stating any required assumptions.
3. Website Speed. The speed at which you can log into a website through a smartphone is an important
quality characteristic of that website. In a recent test, the mean time to log into the INFORMS
OR/MS Tomorrow website was 6.734 seconds. Suppose that the download time is normally distributed, with a standard deviation of 2.7 seconds. What is the probability that a download time
is
(a) (2 points) less than 2 seconds?
(b) (2 points) between 1.8 and 2.4 seconds?
(c) (2 points) 99% of the download times are slower (higher) than how many seconds?
(d) (2 points) 95% of the download times are between what two values, symmetrically distributed
around the mean?
(e) (1 point) Suppose the download times are uniformly distributed between 1 to 13 seconds. What
is your answer to 3(b)?


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