### 统计代写|统计推断代写Statistical inference代考|STAT3023

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|统计推断代写Statistical inference代考|Further exercises

1. Consider a probability space $(\Omega, \mathcal{F}, \mathrm{P})$ and events $A, B, C \in \mathcal{F}$.
(a) If $\mathrm{P}(A)=\frac{3}{4}$ and $\mathrm{P}(B)=\frac{1}{3}$, show that $\frac{1}{12} \leq \mathrm{P}(A \cap B) \leq \frac{1}{3}$. When do the two equalities hold?
(b) Is it possible to find events for which the following four conditions hold: $A \cap B \subset C^{c}, \mathrm{P}(A)>0.5, \mathrm{P}(B)>0.5$, and $\mathrm{P}(C)>0.5$ ?
(c) If $A \cap B \subset C^{c}, \mathrm{P}(A)=0.5$, and $\mathrm{P}(B)=0.5$, what is the largest possible value for $\mathrm{P}(C)$ ?
2. Consider a probability space $(\Omega, \mathcal{F}, \mathrm{P})$ and events $A, B, C \in \mathcal{F}$. Starting from the definition of probability measure, show that
(a) $\mathrm{P}\left((A \cup B) \cap\left(A^{c} \cup B^{c}\right)\right)=\mathrm{P}(A)+\mathrm{P}(B)-2 \mathrm{P}(A \cap B)$.
(b) $\mathrm{P}(A \cup B \cup C)=\mathrm{P}(A)+\mathrm{P}(B)+\mathrm{P}(C)-\mathrm{P}(A \cap B)-\mathrm{P}(A \cap C)-\mathrm{P}(B \cap C)+\mathrm{P}(A \cap B \cap C)$.
3.(a) In 1995 an account on the LSE network came with a three letter (all uppercase Roman letter) password. Suppose a malicious hacker could check one password every millisecond. Assuming the hacker knows a username and the format of passwords, what is the maximum time that it would take to break into an account?
(b) In a bid to improve security, IT services propose to either double the number of letters available (by including lowercase letters) or double the length (from three to six). Which of these options would you recommend? Is there a fundamental principle here that could be applied in other situations?
(c) Suppose that, to be on the safe side, IT services double the number of letters, include numbers, and increase the password length to twelve. You have forgotten your password. You remember that it contains the characters ${t, t, t, S, s, s, I, i, i, c, a, 3}$. If you can check passwords at the same rate as a hacker, how long will it take you to get into your account?
3. $A$ and $B$ are events of positive probability. Supply a proof for each of the following.
(a) If $A$ and $B$ are independent, $A$ and $B^{c}$ are independent.
(b) If $A$ and $B$ are independent, $\mathrm{P}\left(A^{c} \mid B^{c}\right)+\mathrm{P}(A \mid B)=1$.
(c) If $\mathrm{P}(A \mid B)<\mathrm{P}(A)$, then $\mathrm{P}(B \mid A)<\mathrm{P}(B)$.
(d) If $\mathrm{P}(B \mid A)=\mathrm{P}\left(B \mid A^{c}\right)$ then $A$ and $B$ are independent.
4. A fair coin is independently tossed twice. Consider the following events:
$A=$ “The first toss is heads”
$B=$ “The second toss is heads”
$C=$ “First and second toss show the same side”
Show that $A, B, C$ are pairwise independent events, but not independent events.
5. Show that, if $A, B$, and $C$ are independent events with $\mathrm{P}(A)=\mathrm{P}(B)=\mathrm{P}(C)$, then the probability that exactly one of $A, B$, and $C$ occurs is less than or equal to $4 / 9$.
6. Consider a probability space $(\Omega, \mathcal{F}, \mathrm{P})$ and events $A, B, C_{1}, C_{2} \in \mathcal{F}$. Suppose, in addition, that $C_{1} \cap C_{2}=\varnothing$ and $C_{1} \cup C_{2}=B$. Show that
$$\mathrm{P}(A \mid B)=\mathrm{P}\left(A \mid C_{1}\right) \mathrm{P}\left(C_{1} \mid B\right)+\mathrm{P}\left(A \mid C_{2}\right) \mathrm{P}\left(C_{2} \mid B\right) .$$

## 统计代写|统计推断代写Statistical inference代考|Mapping outcomes to real numbers

At an intuitive level, the definition of a random variable is straightforward; a random variable is a quantity whose value is determined by the outcome of the experiment. The value taken by a random variable is always real. The randomness of a random variable is a consequence of our uncertainty about the outcome of the experiment. Example 3.1.1 illustrates this intuitive thinking, using the setup described in Example 2.4.14 as a starting point.

In practice, the quantities we model using random variables may be the output of systems that cannot be viewed as experiments in the strict sense. What these systems have in common, however, is that they are stochastic, rather than deterministic. This is an important distinction; for a deterministic system, if we know the input, we can determine exactly what the output will be. This is not true for a stochastic model, as its output is (at least in part) determined by a random element. We will encounter again the distinction between stochastic and deterministic systems in Chapter 12 , in the context of random-number generation.
Example 3.1.1 (Coin flipping again)
Define a random variable $X$ to be the number of heads when we flip a coin three times. We assume that flips are independent and that the probability of a head at each flip is $p$. We know that $X$ can take one of four values, $0,1,2$, or 3 . For convenience,

we say that $X$ can take any real value, but the probability of it taking a value outside ${0,1,2,3}$ is zero. The probabilities evaluated in Example 2.4.14 can now be written as
$$P(X=x)= \begin{cases}(1-p)^{3}, & x=0 \ 3 p(1-p)^{2}, & x=1 \ 3 p^{2}(1-p), & x=2 \ p^{3}, & x=3 \ 0, & \text { otherwise }\end{cases}$$

## 统计代写|统计推断代写Statistical inference代考|Cumulative distribution functions

As mentioned above, we are usually more interested in probabilities associated with a random variable than in a mapping from outcomes to real numbers. The probability associated with a random variable is completely characterised by its cumulative distribution function.

Definition 3.2.1 (Cumulative distribution function)
The cumulative distribution function (CDF) of a random variable $X$ is the function $F_{X}: \mathbb{R} \longrightarrow[0,1]$ given by $F_{X}(x)=\mathrm{P}(X \leq x)$
A couple of points to note about cumulative distribution functions.

1. We will use $F_{X}$ to denote the cumulative distribution function of the random variable $X, F_{Y}$ to denote the cumulative distribution function of the random variable $Y$, and so on.
2. Be warned; some texts use the argument to identify different distribution functions. For example, you may see $F(x)$ and $F(y)$ used, not to denote the same function applied to different arguments, but to indicate a value of the cumulative distribution function of $X$ and a value of the cumulative distribution function of $Y$. This can be deeply confusing and we will try to avoid doing it.

In our discussion of the properties of cumulative distribution functions, the following definition is useful.
Definition 3.2.2 (Right continuity)
A function $g: \mathbb{R} \rightarrow \mathbb{R}$ is right-continuous if $g(x+)=g(x)$ for all $x \in \mathbb{R}$, where $g(x+)=\lim _{h \downarrow 0} g(x+h)$

The notation $g(x+)$ is used for limit from the right. There is nothing complicated about this; it is just the limit of the values given by $g$ as we approach the point $x$ from the right-hand side. Right continuity says that we can approach any point from the right-hand side without encountering a jump in the value given by $g$. There is an analogous definition of left continuity in terms of the limit from the left; $g$ is left-continuous if $g(x-)=\lim {h \downarrow 0} g(x-h)=g(x)$ for all $x$. Somewhat confusingly, the notation $\lim {h \uparrow 0}$ is sometimes used. This is discussed as part of Exercise 3.2.
The elementary properties of cumulative distribution functions are inherited from their definition in terms of probability. It is true, but rather harder to show, that any function satisfying the three properties given in Proposition $3.2 .3$ is the distribution function of some random variable. We will only prove necessity of the three conditions.

## 统计代写|统计推断代写Statistical inference代考|Further exercises

1. 考虑一个概率空间(Ω,F,磷)和事件一个,乙,C∈F.
(a) 如果磷(一个)=34和磷(乙)=13， 显示112≤磷(一个∩乙)≤13. 这两个等式何时成立？
(b) 是否有可能找到满足以下四个条件的事件：一个∩乙⊂CC,磷(一个)>0.5,磷(乙)>0.5， 和磷(C)>0.5?
(c) 如果一个∩乙⊂CC,磷(一个)=0.5， 和磷(乙)=0.5, 的最大可能值是多少磷(C) ?
2. 考虑一个概率空间(Ω,F,磷)和事件一个,乙,C∈F. 从概率测度的定义出发，证明
(a)磷((一个∪乙)∩(一个C∪乙C))=磷(一个)+磷(乙)−2磷(一个∩乙).
(二)磷(一个∪乙∪C)=磷(一个)+磷(乙)+磷(C)−磷(一个∩乙)−磷(一个∩C)−磷(乙∩C)+磷(一个∩乙∩C).
3.(a) 1995 年，LSE 网络上的一个帐户带有三个字母（全为大写罗马字母）的密码。假设恶意黑客可以每毫秒检查一个密码。假设黑客知道用户名和密码格式，那么入侵帐户所需的最长时间是多少？
(b) 为了提高安全性，IT 服务建议将可用字母的数量增加一倍（包括小写字母）或将长度增加一倍（从 3 个到 6 个）。您会推荐以下哪些选项？这里有一个基本原则可以应用于其他情况吗？
(c) 假设为了安全起见，IT 服务将字母数量加倍，包括数字，并将密码长度增加到 12。您忘记了密码。你记得它包含字符吨,吨,吨,小号,s,s,我,一世,一世,C,一个,3. 如果您可以像黑客一样检查密码，您需要多长时间才能进入您的帐户？
3. 一个和乙是正概率事件。为以下每一项提供证明。
(a) 如果一个和乙是独立的，一个和乙C是独立的。
(b) 如果一个和乙是独立的，磷(一个C∣乙C)+磷(一个∣乙)=1.
(c) 如果磷(一个∣乙)<磷(一个)， 然后磷(乙∣一个)<磷(乙).
(d) 如果磷(乙∣一个)=磷(乙∣一个C)然后一个和乙是独立的。
4. 一枚公平的硬币独立投掷两次。考虑以下事件：
一个=“第一次投掷是正面”
乙=“第二次投掷是正面”
C=“第一次和第二次折腾显示同一面”
表明一个,乙,C是成对独立事件，但不是独立事件。
5. 证明，如果一个,乙， 和C是独立的事件磷(一个)=磷(乙)=磷(C)，那么恰好其中之一的概率一个,乙， 和C发生小于或等于4/9.
6. 考虑一个概率空间(Ω,F,磷)和事件一个,乙,C1,C2∈F. 此外，假设C1∩C2=∅和C1∪C2=乙. 显示
磷(一个∣乙)=磷(一个∣C1)磷(C1∣乙)+磷(一个∣C2)磷(C2∣乙).

## 统计代写|统计推断代写Statistical inference代考|Cumulative distribution functions

1. 我们将使用FX表示随机变量的累积分布函数X,F是表示随机变量的累积分布函数是， 等等。
2. 被警告; 一些文本使用参数来识别不同的分布函数。例如，您可能会看到F(X)和F(是)使用，不是表示应用于不同参数的相同函数，而是表示累积分布函数的值X和累积分布函数的值是. 这可能会让人非常困惑，我们会尽量避免这样做。

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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。