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# 统计代写|统计推断代考Statistical Inference代写|Some Philosophical Applications

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## 统计代写|统计推断代考Statistical Inference代写|Some Philosophical Applications

Streaks
In the previous section we looked at the probability of getting a certain number of heads in a number of flips. Look at the following two sequences:
2 HНТHННТТTТТТТНТНТТНТТТНТНТННТНТТНТТТННТТТННННТНННН
One of these sequences was generated from actually flipping a coin 50 times. The other one is from a person pretending to flip a coin, and writing down a sequence that they thought would look like a random flipping of a coin. Which one is which? While many people think that sequence 1 looks more “random” (i.e. it seems to flip around a lot), sequence 2 is actually the random sequence.

One of the truly unintuitive things about real random sequences, as opposed to designed sequences, is that there are long runs or streaks. Why is this? The general solution is beyond this book but we can think about it this way. Although a sequence of, say, 5 heads in a row is very unlikely $\left(P(5\right.$ heads in a row $\left.)=(1 / 2)^5=0.03\right)$, there are many opportunities for such a sequence somewhere within a sequence of 50 . Because of these many opportunities, this raises the probability from $3 \%$ (the probability of 5 heads in a row in 5 flips), to over $55 \%$, the probability of finding 5 heads in a row somewhere in 50 flips. Streaks of 6 heads in a row occur nearly one third of the time in 50 flips, or over half the time if you consider a run to be either heads or tails. Even streaks of 9 heads or tails in a row, in 5 o flips, are not extremely unlikely!
Gambler’s Fallacy
When we look at a sequence of real coin flips, like:

• HHTHHHTTTTTTT
and we ask about the probability of flipping heads in the next flip, it is common to (mistakenly!) reason that, because we’ve seen 7 tails in
• a row, then the next flip is more likely to be heads. However, this is not the case for two reasons:
• 1 long streaks are common in completely fair and random sequences – so observing a streak of 7 tails does not contribute much to one’s confidence that we are looking at a rigged coin or one that has changed its probability properties.
• 2 the process of flipping a coin is independent each time, nearly by definition, and thus the result of one flip cannot influence the result of the next flip. ${ }^1$

## 统计代写|统计推断代考Statistical Inference代写|The Hot Hand – Correlations in Random Sequences

Some work by Tversky and Gilovich ${ }^2$ looks at the following issue in the sport of basketball: there are times when it seems as if basketball players have a “hot hand” – they are on a shooting streak. Tversky and Gilovich looked at how basketball fans perceived streaks, by having them rate sequences of shots as random shooting or streak shooting. Most $(65 \%)$ of the respondents classified artificially generated, purely random sequences as streak shooting. In real data, they discovered that the actual probability of “making a given shot (i.e. a player’s shooting percentage) is unaffected by the player’s prior performance.” We examine this effect in a later section (see Example 9.11 on page 172) where we explore the quantitative procedure for assessing this conclusion. It is enough here to note the large difference between the perception of the sequence and the likely cause of the sequence, and thus the need to always be vigilant against faulty perceptions. Tversky and Gilovich insist that “their observations do not tell us anything general about sports, but it does suggest a generalization about people, namely that they tend to ‘detect’ patterns even where none exist.”

What we have here, again, is the general perception that long sequences are somehow not “random,” when in fact the opposite is the case. People have a natural tendency to see patterns in random data, to infer order where there is none, and to ascribe importance to the appearance of pattern. It is the role of statistical inference in general to provide the tools to properly handle the distinction between random effects and patterns, and to retune our intuitions.
Regression Toward the Mean
There is a peculiar phenomenon referred to as regression toward the mean, which often is misinterpreted and leads to failures of proper statistical inference. It can be seen in a simple example. Imagine that we “test” a number of students by having them guess the results of a coin flip. Clearly this will be entirely luck, because the coin flip has no pattern. If a student guesses the results of 50 flips, there will be an expectation of getting 25 correct. Here we simulate 20 students each “predicting” the result of 50 flips, the results shown in Table 3.1. The test is done twice, and we will look at a particular subset presently. One can, by eye, see that most of the students get around 25 correct exactly as expected from random performance.

Now, imagine that we look at the top five coin flip predictors on the first round. Will they do better or worse in the the second round? What about the bottom five coin flip predictors? The results of these two cases are summarized in Table 3.2. The pattern, even in this small sample, is quite clear:
1 Those that did the best the first time did worse the second (on average)
2 Those that did the worst the first time did better the second (on average)

# 统计推断代写

## 统计代写|统计推断代考Statistical Inference代写|Some Philosophical Applications

2 HНТHННТТTТТТТНТНТТНТТТНТНТННТНТТНТТТННТТТННННТНННН

HHTHHHTTTTTTT

## 统计代写|统计推断代考Statistical Inference代写|The Hot Hand – Correlations in Random Sequences

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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。