## 澳洲代写｜PHYC30018｜Quantum Physics量子计算 墨尔本大学

statistics-labTM为您提供墨尔本大学The University of Melbourne，简称UniMelb，中文简称“墨大”）Quantum Physics量子计算澳洲代写代考辅导服务！

Quantum mechanics plays a central role in our understanding of fundamental phenomena, primarily in the microscopic domain. It lays the foundation for an understanding of atomic, molecular, condensed matter, nuclear and particle physics.

## Quantum Physics量子计算问题集

(a) Suppose that the resistivity matrix is given by the classical result
$$\rho=\left(\begin{array}{cc} \rho_0 & -\rho_H \ \rho_H & \rho_0 \end{array}\right)$$
where $\rho_H=B /$ nec is the Hall resistivity and $\rho_0$ is the usual Ohmic resistivity. Find the conductivity matrix, $\sigma=\rho^{-1}$. Write it in the form:
$$\sigma=\left(\begin{array}{cc} \sigma_0 & \sigma_H \ -\sigma_H & \sigma_0 \end{array}\right) .$$
What are $\sigma_0$ and $\sigma_H$ ?
(b) Suppose $B=0$, so the Hall resistivity is zero. Notice that the Ohmic conductivity, $\sigma_0$, is just $1 / \rho_0$. In particular, note that $\sigma_0 \rightarrow \infty$ as $\rho_0 \rightarrow 0$. Now suppose $\rho_H \neq 0$. Show that $\sigma_0 \rightarrow 0$ as $\rho_0 \rightarrow 0$, so it is possible to have both $\sigma_0$ and $\rho_0$ equal to zero

This problem asks you to give a complete presentation of a calculation that is almost the same as one you saw in lecture.

Consider a constant electric field, $\vec{E}=\left(0, E_0, 0\right)$ and a constant magnetic field, $\vec{B}=\left(0,0, B_0\right)$.
(a) Choose an electrostatic potential $\phi$ and a vector potential $\vec{A}$ which describe the $\vec{E}$ and $\vec{B}$ fields, and write the Hamiltonian for a charged particle of mass $m$ and charge $q$ in these fields. Assume that the particle is restricted to move in the $x y$-plane.
(b) What are the allowed energies as a function of $B_0$ and $E_0$ ? Draw a figure to show how the Landau levels (energy levels when $E_0=0$ ) change as $E_0$ increases.

You will see the “standard presentation” of the Aharonov-Bohm effect in lecture, on the day that this problem set is due. The standard presentation has its advantages, and in particular is more general than the presentation you will work through in this problem. However, students often come away from the standard presentation of the Aharonov-Bohm effect thinking that the only way to detect this effect is to do an interference experiment. This is not true, and the purpose of this problem is to disabuse you of this misimpression before you form it.

As Griffiths explains on pages 385-387 (344-345 in 1st Ed.), the Aharonov-Bohm effect modifies the energy eigenvalues of suitably chosen quantum mechanical systems. In this problem, I ask you to work through the same physical example that Griffiths uses, but in a different fashion which makes more use of gauge invariance.

Imagine a particle constrained to move on a circle of radius $b$ (a bead on a wire ring, if you like.) Along the axis of the circle runs a solenoid of radius $a<b$, carrying a magnetic field $\vec{B}=\left(0,0, B_0\right)$. The field inside the solenoid is uniform

and the field outside the solenoid is zero. The setup is depicted in Griffiths’ Fig. 10.10. (10.12 in 1st Ed.)
(a) Construct a vector potential $\vec{A}$ which describes the magnetic field (both inside and outside the solenoid) and which has the form $A_r=A_z=0$ and $A_\phi=\alpha(r)$ for some function $\alpha(r)$. I am using cylindrical coordinates $z, r$, $\phi$.
(b) Since $\vec{\nabla} \times \vec{A}=0$ for $r>a$, it must be possible to write $\vec{A}=\vec{\nabla} f$ in any simply connected region in $r>a$. [This is a theorem in vector calculus.] Show that if we find such an $f$ in the region
$$r>a \text { and }-\pi+\epsilon<\phi<\pi-\epsilon,$$
then
$$f(r, \pi-\epsilon)-f(r,-\pi+\epsilon) \rightarrow \Phi \text { as } \epsilon \rightarrow 0 .$$
Here, the total magnetic flux is $\Phi=\pi a^2 B_0$. Now find an explicit form for $f$, which is a function only of $\phi$.
(c) Now consider the motion of a “bead on a ring”: write the Schrödinger equation for the particle constrained to move on the circle $r=b$, using the $\vec{A}$ you found in (a). Hint: the answer is given in Griffiths.
(d) Use the $f(\phi)$ found in (b) to gauge transform the Schrödinger equation for $\psi(\phi)$ within the angular region $-\pi+\epsilon<\phi<\pi-\epsilon$ to a Schrödinger equation for a free particle within this angular region. Call the original wave function $\psi(\phi)$ and the gauge-transformed wave function $\psi^{\prime}(\phi)$.
(e) The original wave function $\psi$ must be single-valued for all $\phi$, in particular at $\phi=\pi$. That is, $\psi(\pi-\epsilon)-\psi(-\pi+\epsilon) \rightarrow 0$ and $\frac{\partial \psi}{\partial \phi}(\pi-\epsilon)-\frac{\partial \psi}{\partial \phi}(-\pi+\epsilon) \rightarrow 0$ as $\epsilon \rightarrow 0$. What does this say about the gauge-transformed wave function? I.e., how must $\psi^{\prime}(\pi-\epsilon)$ and $\psi^{\prime}(-\pi+\epsilon)$ be related as $\epsilon \rightarrow 0$ ?
[Hint: because the $f(\phi)$ is not single valued at $\phi=\pi$, the gauge transformed wave function $\psi^{\prime}(\phi)$ is not single valued there either.]
(f) Solve the Schrödinger equation for $\psi^{\prime}$, and find energy eigenstates which satisfy the boundary conditions you derived in (e). What are the allowed energy eigenvalues?
(g) Undo the gauge transformation, and find the energy eigenstates $\psi(\phi)$ in the original gauge. Do the energy eigenvalues in the two gauges differ?
(h) Plot the energy eigenvalues as a function of the enclosed flux, $\Phi$. Show that the energy eigenvalues are periodic functions of $\Phi$ with period $\Phi_0$, where you must determine $\Phi_0$. For what values of $\Phi$ does the enclosed magnetic field have no effect on the spectrum of a particle on a ring? Show that the

Aharonov-Bohm effect can only be used to determine the fractional part of $\Phi / \Phi_0$.
[Note: you have shown that even though the bead on a ring is everywhere in a region in which $\vec{B}=0$, the presence of a nonzero $\vec{A}$ affects the energy eigenvalue spectrum. However, the effect on the energy eigenvalues is determined by $\Phi$, and is therefore gauge invariant. To confirm the gauge invariance of your result, you can compare your answer for the energy eigenvalues to Griffiths’ result, obtained using a different gauge.]

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 澳洲代写｜COMP30027｜Machine Learning机器学习 墨尔本大学

statistics-labTM为您提供墨尔本大学The University of Melbourne，简称UniMelb，中文简称“墨大”）Complex Analysis复杂分析澳洲代写代考辅导服务！

Machine Learning, a core discipline in data science, is prevalent across Science, Technology, the Social Sciences, and Medicine; it drives many of the products we use daily such as banner ad selection, email spam filtering, and social media newsfeeds. Machine Learning is concerned with making accurate, computationally efficient, interpretable and robust inferences from data. Originally borne out of Artificial Intelligence, Machine Learning has historically been the first to explore more complex prediction models and to emphasise computation, while in the past two decades Machine Learning has grown closer to Statistics gaining firm theoretical footing.

## Machine Learning机器学习 问题集

The data and scripts for this problem are available in hw2/prob1. You can load the data using the MATLAB script load_al_data. This script should load the matrices y_noisy, y_true, X_in. The $y$ vectors are $n \times 1$ while $\mathrm{X}{-}$in is a $n \times 3$ matrix with each row corresponding to a point in $\mathcal{R}^3$. The $y{\text {true }}$ vectors correspond to the ideal $y$ values, generated directly from the “true” model (whatever it may be) without any noise. In contrast, the $y_{\text {noisy }}$ vectors are the actual, noisy observations, generated by adding Gaussian noise to the $y_{\text {true }}$ vectors. You should use $y_{n o i s y}$ for any estimation. $y_{\text {true }}$ is provided only to make it easier to evaluate the error in your predictions (simulate an infinite test data). You would not have $y_{\text {true }}$ in any real task.
(a) Write MATLAB functions theta = linear_regress $(\mathrm{y}, \mathrm{X})$ and $\mathrm{y}$ hat $=$ linear_pred(theta, X_test). Note that we are not explicitly including the offset parameter but instead rely on the feature vectors to provide a constant component. See part (b).
(b) The feature mapping can substantially affect the regression results. We will consider two possible feature mappings:
\begin{aligned} & \phi_1\left(x_1, x_2, x_3\right)=\left[1, x_1, x_2, x_3\right]^T \ & \phi_2\left(x_1, x_2, x_3\right)=\left[1, \log x_1^2, \log x_2^2, \log x_3^2\right]^T \end{aligned}
Use the provided MATLAB function feature mapping to transform the input data matrix into a matrix
$$X=\left[\begin{array}{c} \phi\left(\mathbf{x}1\right)^T \ \phi\left(\mathbf{x}_2\right)^T \ \cdots \ \phi\left(\mathbf{x}{\mathbf{n}}\right)^T \end{array}\right]$$
For example, $\mathrm{X}$ = feature_mapping ( $\mathrm{X}_{-}$in, 1 ) would get you the first feature representation. Using your completed linear regression functions, compute the mean squared prediction error for each feature mapping (2 numbers).

(c) The selection of points to query in an active learning framework might depend on the feature representation. We will use the same selection criterion as in the lectures, the expected squared error in the parameters, proportional to $\operatorname{Tr}\left[\left(X^T X\right)^{-1}\right]$. Write a MATLAB function $\mathrm{idx}=\operatorname{active_ learn}(\mathrm{X}, \mathrm{k} 1, \mathrm{k} 2)$. Your function should assume that the top $k_1$ rows in $X$ have been queried and your goal is to sequentially find the indices of the next $k_2$ points to query. The final set of $k_1+k_2$ indices should be returned in idx. The latter may contain repeated entries. For each feature mapping, and $k_1=5$ and $k_2=10$, compute the set of points that should be queried (i.e., $\mathrm{X}(:, \mathrm{idx})$ ). For each set of points, use the feature mapping $\phi_2$ to perform regression and compute the resulting mean squared prediction errors (MSE) over the entire data set (again, using $\phi_2$ ).

(d) Let us repeat the steps of part (c) with randomly selected additional points to query. We have provided a MATLAB function $i d x=\operatorname{randomly}(\operatorname{select}(\mathrm{X}, \mathrm{k} 1, \mathrm{k} 2)$ which is essentially the same as active_learn except that it selects the $k_2$ points uniformly at random from $X$. Repeat the regression steps as in previous part, and compute the resulting mean squared prediction error again. To get a reasonable comparison you should repeat this process 50 times, and use the median MSE. Compare the resulting errors with the active learning strategies. What conclusions can you draw?

(e) Let us now compare the two sets of points chosen by active learning due to the different feature representations. We have provided a function plot_points(X,idx_r,idx_b) which will plot each row of $X$ as a point in $\mathbf{R}^3$. The points indexed by $i d x _r$ will be circled in red and those marked by idx_b will be circled (larger) in blue (some of the points indexed by idx_r and idx_b might be common). Plot the original data points using the indexes of the actively selected points based on the two feature representations. Also plot the same indexes using $\mathrm{X}$ from the second feature representation with its first constant column removed. In class, we saw an example where the active learning strategy chose points at the extrema of the available space. Can you see evidence of this in the two plots?

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 澳洲代写｜MAST90104｜A First Course In Statistical Learning 统计学习第一门课程 墨尔本大学

statistics-labTM为您提供墨尔本大学The University of Melbourne，简称UniMelb，中文简称“墨大”）A First Course In Statistical Learning 统计学习第一门课程澳洲代写代考辅导服务！

Supervised statistical learning is based on the widely used linear models that model a response as a linear combination of explanatory variables. Initially this subject develops an elegant unified theory for a quantitative response that includes the estimation of model parameters, hypothesis testing using analysis of variance, model selection, diagnostics on model assumptions, and prediction. Some classification methods for qualitative responses are then developed. This subject then considers computational techniques, including the EM algorithm. Bayes methods and Monte-Carlo methods are considered. The subject concludes by considering some unsupervised learning techniques.

## A First Course In Statistical Learning 统计学习第一门课程案例

Describe the unethical behavior in each example and describe how it could impact the reliability of the resulting data. Explain how the problem should be corrected.
A researcher is collecting data in a community.
a. She selects a block where she is comfortable walking because she knows many of the people living on the street.
b. No one seems to be home at four houses on her route. She does not record the addresses and does not return at a later time to try to find residents at home.
c. She skips four houses on her route because she is running late for an appointment. When she gets home, she fills in the forms by selecting random answers from other residents in the neighborhood.

a. By selecting a convenient sample, the researcher is intentionally selecting a sample that could be biased. Claiming that this sample represents the community is misleading. The researcher needs to select areas in the community at random.
b. Intentionally omitting relevant data will create bias in the sample. Suppose the researcher is gathering information about jobs and child care. By ignoring people who are not home, she may be missing data from working families that are relevant to her study. She needs to make every effort to interview all members of the target sample.
c. It is never acceptable to fake data. Even though the responses she uses are “real” responses provided by other participants, the duplication is fraudulent and can create bias in the data. She needs to work diligently to interview everyone on her route.

Listed are 29 ages for Academy Award winning best actors in order from smallest to largest.
$$18 ; 21 ; 22 ; 25 ; 26 ; 27 ; 29 ; 30 ; 31 ; 33 ; 36 ; 37 ; 41 ; 42 ; 47 ; 52 ; 55 ; 57 ; 58 ; 62 ; 64 ; 67 ; 69 ; 71 ; 72 ; 73 ; 74 ; 76 ; 77$$
a. Find the percentile for 58 .
b. Find the percentile for 25 .

a. Counting from the bottom of the list, there are 18 data values less than 58 . There is one value of 58 . $x=18$ and $y=1 . \frac{x+0.5 y}{n}(100)=\frac{18+0.5(1)}{29}(100)=63.80 .58$ is the $64^{\text {th }}$ percentile.
b. Counting from the bottom of the list, there are three data values less than 25 . There is one value of 25 . $x=3$ and $y=1 . \frac{x+0.5 y}{n}(100)=\frac{3+0.5(1)}{29}(100)=12.07$. Twenty-five is the $12^{\text {th }}$ percentile.

Suppose that in a small town of 50 people, one person earns $\$ 5,000,000$per year and the other 49 each earn$\$30,000$. Which is the better measure of the “center”: the mean or the median?

\begin{aligned} \bar{x} & =\frac{5,000,000+49(30,000)}{50}=129,400 \ M & =30,000 \end{aligned}
(There are 49 people who earn $\$ 30,000$and one person who earns$\$5,000,000$.)
The median is a better measure of the “center” than the mean because 49 of the values are 30,000 and one is $5,000,000$. The $5,000,000$ is an outlier. The 30,000 gives us a better sense of the middle of the data.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 澳洲代写｜MAST30020｜Probability for Inference推理概率 墨尔本大学

statistics-labTM为您提供墨尔本大学The University of Melbourne，简称UniMelb，中文简称“墨大”）Probability for Inference推理概率要素澳洲代写代考辅导服务！

This subject introduces a measured-theoretic approach to probability theory and presents its fundamentals concepts and results.

Topics covered include: probability spaces and random variables, expectation, conditional expectation and distributions, elements of multivariate distribution theory, modes of convergence in probabilty theory, characteristics functions and their application in key limit theorems.

## Probability for Inference推理概率案例

A survey was taken of a group’s viewing habits of sporting events on TV during the last year. Let $A={$ watched football $}, B={$ watched basketball $}, C={$ watched baseball $}$. The results indicate that if a person is selected at random from the surveyed group, then $P(A)=0.43, P(B)=0.40, P(C)=0.32, P(A \cap B)=0.29$, $P(A \cap C)=0.22, P(B \cap C)=0.20$, and $P(A \cap B \cap C)=0.15$. It then follows that
\begin{aligned} P(A \cup B \cup C)= & P(A)+P(B)+P(C)-P(A \cap B)-P(A \cap C) \ & -P(B \cap C)+P(A \cap B \cap C) \ = & 0.43+0.40+0.32-0.29-0.22-0.20+0.15 \ = & 0.59 \end{aligned}
is the probability that this person watched at least one of these sports.
Let a probability set function be defined on a sample space $S$. Let $S=\left{e_1, e_2, \ldots, e_m\right}$, where each $e_i$ is a possible outcome of the experiment. The integer $m$ is called the total number of ways in which the random experiment can terminate. If each of these outcomes has the same probability of occurring, we say that the $m$ outcomes are equally likely. That is,
$$P\left(\left{e_i\right}\right)=\frac{1}{m}, \quad i=1,2, \ldots, m .$$
If the number of outcomes in an event $A$ is $h$, then the integer $h$ is called the number of ways that are favorable to the event $A$. In this case, $P(A)$ is equal to the number of ways favorable to the event $A$ divided by the total number of ways in which the experiment can terminate. That is, under this assumption of equally likely outcomes, we have
$$P(A)=\frac{h}{m}=\frac{N(A)}{N(S)},$$
where $h=N(A)$ is the number of ways $A$ can occur and $m=N(S)$ is the number of ways $S$ can occur. Exercise 1.1-15 considers this assignment of probability in a more theoretical manner.

It should be emphasized that in order to assign the probability $h / m$ to the event $A$, we must assume that each of the outcomes $e_1, e_2, \ldots, e_m$ has the same probability $1 / \mathrm{m}$. This assumption is then an important part of our probability model; if it is not realistic in an application, then the probability of the event $A$ cannot be computed in this way. Actually, we have used this result in the simple case given in Example 1.1-3 because it seemed realistic to assume that each of the possible outcomes in $S={H H, H T, T H, T T}$ had the same chance of being observed.

\begin{aligned} P(A\cap B\cap C)= & P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)\ & -P(B \cap C)+P(A \cap B \cap C) \ = & 0.43+0.40+0.32-0.29-0.22-0.20+0.15 \ = & 0.59 \end{aligned}

$$P\left(\left{e_i\right}\right)=\frac{1}{m}, \quad i=1,2, \ldots, m .$$

$$P(A)=\frac{h}{m}=\frac{N(A)}{N(S)},$$

## Probability for Inference推理概率 案例2

Let the random experiment be the cast of a die. Then the outcome space associated with this experiment is $S={1,2,3,4,5,6}$, with the elements of $S$ indicating the number of spots on the side facing up. For each $s \in S$, let $X(s)=s$. The space of the random variable $X$ is then ${1,2,3,4,5,6}$.

If we associate a probability of $1 / 6$ with each outcome, then, for example, $P(X=5)=1 / 6, P(2 \leq X \leq 5)=4 / 6$, and $P(X \leq 2)=2 / 6$ seem to be reasonable assignments, where, in this example, ${2 \leq X \leq 5}$ means ${X=2,3,4$, or 5$}$ and ${X \leq 2}$ means ${X=1$ or 2$}$.
The student will no doubt recognize two major difficulties here:

1. In many practical situations, the probabilities assigned to the events are unknown.
2. Since there are many ways of defining a function $X$ on $S$, which function do we want to use?

As a matter of fact, the solutions to these problems in particular cases are major concerns in applied statistics. In considering (2), statisticians try to determine what measurement (or measurements) should be taken on an outcome; that is, how best do we “mathematize” the outcome? These measurement problems are most difficult and can be answered only by getting involved in a practical project. For (1), we often need to estimate these probabilities or percentages through repeated observations (called sampling). For example, what percentage of newborn girls in the University of Iowa Hospital weigh less than 7 pounds? Here a newborn baby girl is the outcome, and we have measured her one way (by weight), but obviously there are many other ways of measuring her. If we let $X$ be the weight in pounds, we are interested in the probability $P(X<7)$, and we can estimate this probability only by repeated observations. One obvious way of estimating it is by the use of the relative frequency of ${X<7}$ after a number of observations. If it is reasonable to make additional assumptions, we will study other ways of estimating that probability. It is this latter aspect with which the field of mathematical statistics is concerned. That is, if we assume certain models, we find that the theory of statistics can explain how best to draw conclusions or make predictions.

In many instances, it is clear exactly what function $X$ the experimenter wants to define on the outcome space. For example, the caster in the dice game called craps is concerned about the sum of the spots (say $X$ ) that are facing upward on the pair of dice. Hence, we go directly to the space of $X$, which we shall denote by the same letter $S$. After all, in the dice game the caster is directly concerned only with the probabilities associated with $X$. Thus, for convenience, in many instances the reader can think of the space of $X$ as being the outcome space.

Let $X$ denote a random variable with space $S$. Suppose that we know how the probability is distributed over the various subsets $A$ of $S$; that is, we can compute $P(X \in A)$. In this sense, we speak of the distribution of the random variable $X$, meaning, of course, the distribution of probability associated with the space $S$ of $X$.

1. 在许多实际情况中，事件的概率是未知的。
2. 既然有许多方法可以定义 $S$ 上的函数 $X$，那么我们要使用哪个函数呢？

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