### 统计代写|统计推断代写Statistical inference代考|STAT3013

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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|Parameters and families of distributions

A parameter is a characteristic of a distribution that is of interest. Examples of parameters are the probability of success, $p$, for a binomial distribution and the mean, $\mu$, of a normal. Parameters often arise as terms in mass or density functions; the parameter, $\lambda$, of an exponential distribution determines the rate at which the density converges to zero. A distribution family is a set of distributions that differ only in the value of their parameters. For example, consider the number of heads when flipping a fair coin once (can be 0 or 1 ) and the number of sixes when throwing a fair die once (can also be 0 or 1 ).

Exercise $3.3$

1. Show that $\Gamma(1 / 2)=\sqrt{\pi}$, and hence write down a formula for $\Gamma(k / 2)$, where $k$ is a positive integer. [Hint: In the definition of $\Gamma(1 / 2)$, use the substitution $z=\sqrt{2 u}$ and rearrange the integrand to obtain a standard normal density function.]
2. (Geometric mass function) Show that the function
$$f_{X}(x)=(1-p)^{x-1} p \text { for } x=1,2, \ldots$$
is a valid mass function.
3. (Geometric cumulative distribution function) Suppose that $X \sim \operatorname{Geometric}(p)$. Find the cumulative distribution function of $X$.
4. (Negative binomial mass function) Show that the function
$$f_{X}(x)=\left(\begin{array}{l} x-1 \ r-1 \end{array}\right) p^{r}(1-p)^{x-r} \text { for } x=r, r+1, \ldots$$
is a valid mass function.
5. (Cumulative distribution function of a continuous uniform) Suppose that $X \sim$ Unif $[a, b]$. Derive the cumulative distribution function of $X$.
6. Show that the function
$$f_{X}(x)= \begin{cases}\frac{3}{2} x^{2}+x & \text { for } 0 \leq x \leq 1 \ 0 & \text { otherwise }\end{cases}$$
is a valid density function.

## 统计代写|统计推断代写Statistical inference代考|Mean of a random variable

Central tendency is among the first concepts taught on any course in descriptive statistics. The hope is that calculating central tendency will provide us with some sense of the usual or average values taken by an observed variable. Among sample statistics commonly considered are the mode (most commonly occurring value), the median (middle value when observations are ordered) and the arithmetic mean. If we have a massless ruler with points of equal mass placed at locations corresponding to the observed values, the arithmetic mean is the point where we should place a fulcrum in order for the ruler to balance. We will follow the usual convention and refer to the arithmetic mean as just the mean.

These ideas transfer neatly to describing features of distributions. The measures of central tendency that are applied to describe data can also be applied to our models. For example, suppose that $X$ is a continuous random variable with density $f_{X}$ and cumulative distribution function $F_{X}$. We define $\operatorname{mode}(X)=\arg \max {X} f{X}(x)$ and median $(X)=m$, where $m$ is the value satisfying $F_{X}(m)=0.5$. We will now focus our attention on the mean.

## 统计代写|统计推断代写Statistical inference代考|Variance of a random variable

If measures of central tendency are the first thing taught in a course about descriptive statistics, then measures of spread are probably the second. One possible measure of spread is the interquartile range; this is the distance between the point that has a quarter of the probability below it and the point that has a quarter of the probability above it, $\operatorname{IQR}(X)=F_{X}^{-1}(0.75)-F_{X}^{-1}(0.25)$. We will focus on the variance. The variance measures the average squared distance from the mean.
Definition 3.4.7 (Variance and standard deviation)
If $X$ is a random variable, the variance of $X$ is defined as
\begin{aligned} \sigma^{2} &=\operatorname{Var}(X)=\mathbb{B}\left[(X-\mathbb{E}(X))^{2}\right] \ &= \begin{cases}\sum_{x}(x-\mathbb{E}(X))^{2} f_{X}(x) & \text { if } X \text { discrete } \ \int_{-\infty}^{\infty}(x-\mathbb{E}(X))^{2} f_{X}(x) d x & \text { if } X \text { continuous, }\end{cases} \end{aligned}
whenever this sum/integral is finite. The standard deviation is defined as $\sigma=$ $\sqrt{\operatorname{Var}(X)}$
Some properties of the variance operator are given by the following proposition.
Proposition 3.4.8 (Properties of variance)
For a random variable $X$ and real constants $a_{0}$ and $a_{1}$, the variance has the following properties:
i. $\operatorname{Var}(X) \geq 0$,
ii. $\operatorname{Var}\left(a_{0}+a_{1} X\right)=a_{1}^{2} \operatorname{Var}(X)$.
Proof.
Both properties are inherited from the definition of variance as an expectation.
i. By definition, $(X-\mathbb{B}(X))^{2}$ is a positive random variable, so $\operatorname{Var}(X)=\mathbb{B}[(X-$ $\left.\mathbb{B}(X))^{2}\right] \geq 0$ by Claim 3.4.6.
ii. If we define $Y=a_{0}+a_{1} X$, then $\mathbb{E}(Y)=a_{0}+a_{1} \mathbb{E}(X)$, by linearity of expectation. Thus $Y-\mathbb{E}(Y)=a_{1}(X-\mathbb{B}(X))$ and so
\begin{aligned} \operatorname{Var}\left(a_{0}+a_{1} X\right) &=\operatorname{Var}(Y)=\mathbb{E}\left[(Y-\mathbb{B}(Y))^{2}\right]=\mathbb{E}\left[a_{1}^{2}(X-\mathbb{E}(X))^{2}\right] \ &=a_{1}^{2} \operatorname{Var}(X) . \end{aligned}

## 统计代写|统计推断代写Statistical inference代考|Parameters and families of distributions

1. 显示Γ(1/2)=圆周率，因此写下一个公式Γ(ķ/2)， 在哪里ķ是一个正整数。[提示：在定义中Γ(1/2), 使用替换和=2在并重新排列被积函数以获得标准的正态密度函数。]
2. （几何质量函数）显示函数
FX(X)=(1−p)X−1p 为了 X=1,2,…
是一个有效的质量函数。
3. （几何累积分布函数）假设X∼几何的⁡(p). 求累积分布函数X.
4. （负二项式质量函数）证明函数
FX(X)=(X−1 r−1)pr(1−p)X−r 为了 X=r,r+1,…
是一个有效的质量函数。
5. （连续均匀的累积分布函数）假设X∼统一[一个,b]. 导出的累积分布函数X.
6. 显示该函数
FX(X)={32X2+X 为了 0≤X≤1 0 否则
是一个有效的密度函数。

## 统计代写|统计推断代写Statistical inference代考|Variance of a random variable

σ2=曾是⁡(X)=乙[(X−和(X))2] ={∑X(X−和(X))2FX(X) 如果 X 离散的  ∫−∞∞(X−和(X))2FX(X)dX 如果 X 连续的，

i。曾是⁡(X)≥0,

ii. 如果我们定义是=一个0+一个1X， 然后和(是)=一个0+一个1和(X)，通过期望的线性。因此是−和(是)=一个1(X−乙(X))所以

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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。