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# 统计代写|贝叶斯分析代考Bayesian Analysis代写|Example of probability assignment: football point spreads

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## 统计代写|贝叶斯分析代考Bayesian Analysis代写|Example of probability assignment: football point spreads

As an example of how probabilities might be assigned using empirical data and plausible substantive assumptions, we consider methods of estimating the probabilities of certain outcomes in professional (American) football games. This is an example only of probability assignment, not of Bayesian inference. A number of approaches to assigning probabilities for football game outcomes are illustrated: making subjective assessments, using empirical probabilities based on observed data, and constructing a parametric probability model.

Football point spreads and game outcomes
Football experts provide a point spread for every football game as a measure of the difference in ability between the two teams. For example, team A might be a 3.5-point favorite to defeat team B. The implication of this point spread is that the proposition that team A, the favorite, defeats team $B$, the underdog, by 4 or more points is considered a fair bet; in other words, the probability that A wins by more than 3.5 points is $\frac{1}{2}$. If the point spread is an integer, then the implication is that team $\mathrm{A}$ is as likely to win by more points than the point spread as it is to win by fewer points than the point spread (or to lose); there is positive probability that A will win by exactly the point spread, in which case neither side is paid off. The assignment of point spreads is itself an interesting exercise in probabilistic reasoning; one interpretation is that the point spread is the median of the distribution of the gambling population’s beliefs about the possible outcomes of the game. For the rest of this example, we treat point spreads as given and do not worry about how they were derived.

The point spread and actual game outcome for 672 professional football games played during the 1981, 1983, and 1984 seasons are graphed in Figure 1.1. (Much of the 1982 season was canceled due to a labor dispute.) Each point in the scatterplot displays the point spread, $x$, and the actual outcome (favorite’s score minus underdog’s score), $y$. (In games with a point spread of zero, the labels ‘favorite’ and ‘underdog’ were assigned at random.) A small random jitter is added to the $x$ and $y$ coordinate of each point on the graph so that multiple points do not fall exactly on top of each other.

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|Assigning probabilities based on observed frequencies

It is of interest to assign probabilities to particular events: $\operatorname{Pr}$ (favorite wins), $\operatorname{Pr}$ (favorite wins $\mid$ point spread is 3.5 points), $\operatorname{Pr}$ (favorite wins by more than the point spread), $\operatorname{Pr}$ (favorite wins by more than the point spread | point spread is 3.5 points), and so forth. We might report a subjective probability based on informal experience gathered by reading the newspaper and watching football games. The probability that the favored team wins a game should certainly be greater than 0.5 , perhaps between 0.6 and 0.75 ? More complex events require more intuition or knowledge on our part. A more systematic approach is to assign probabilities based on the data in Figure 1.1. Counting a tied game as one-half win and one-half loss, and ignoring games for which the point spread is zero (and thus there is no favorite), we obtain empirical estimates such as:

• $\operatorname{Pr}($ favorite wins $)=\frac{410.5}{655}=0.63$
• $\operatorname{Pr}($ favorite wins $\mid x=3.5)=\frac{36}{59}=0.61$
• $\operatorname{Pr}($ favorite wins by more than the point spread $)=\frac{308}{655}=0.47$
• $\operatorname{Pr}($ favorite wins by more than the point spread $\mid x=3.5)=\frac{32}{59}=0.54$.
These empirical probability assignments all seem sensible in that they match the intuition of knowledgeable football fans. However, such probability assignments are problematic for events with few directly relevant data points. For example, 8.5-point favorites won five out of five times during this three-year period, whereas 9-point favorites won thirteen out of twenty times. However, we realistically expect the probability of winning to be greater for a 9-point favorite than for an 8.5-point favorite. The small sample size with point spread 8.5 leads to imprecise probability assignments. We consider an alternative method using a parametric model.

# 贝叶斯分析代写

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|Example of probability assignment: football point spreads

1981、1983和1984赛季的672场职业足球比赛的分差和实际比赛结果如图1.1所示。(由于劳资纠纷，1982年的大部分节目都被取消了。)散点图中的每个点都显示了点差($x$)和实际结果(最受欢迎的比分减去不受欢迎的比分)$y$。(在分差为0的游戏中，“最受欢迎”和“不受欢迎”的标签是随机分配的。)在图上每个点的$x$和$y$坐标上添加一个小的随机抖动，这样多个点就不会完全落在彼此的顶部。

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|Assigning probabilities based on observed frequencies

$\operatorname{Pr}($ 热门胜利 $)=\frac{410.5}{655}=0.63$

$\operatorname{Pr}($ 热门胜利 $\mid x=3.5)=\frac{36}{59}=0.61$

$\operatorname{Pr}($ 热门队以超过分差的优势获胜 $)=\frac{308}{655}=0.47$

$\operatorname{Pr}($ 夺冠热门以超过分差$\mid x=3.5)=\frac{32}{59}=0.54$获胜。

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。