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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference相关的作业也就用不着说。

• 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|Continuous random variables and density functions

A continuous random variable can be defined as a variable whose cumulative distribution function is continuous. We use a stronger definition that ensures that a continuous random variable has a well-defined (probability) density function.
Definition 3.3.14 (Continuous random variable)
A random variable $X$ is continuous if its distribution function can be expressed as
$$F_X(x)=\int_{-\infty}^x f_X(u) d u \quad \text { for } x \in \mathbb{R}$$
for some integrable function $f_X: \mathbb{R} \longrightarrow[0, \infty)$. The function $f_X$ is called the (probability) density function of $X$.

Definition 3.3.14 shows how the distribution function for a random variable can be found by integrating the density. We can also work in the opposite direction: the density is found by differentiating the cumulative distribution function.
Claim 3.3.15 (Density from cumulative distribution function)
For a continuous random variable $X$ with cumulative distribution function $F_X$, the density function is given by
$$f_X(x)=\left.\frac{d}{d u} F_X(u)\right|{u=x}=F_X^{\prime}(x) \text { for all } x \in \mathbb{R} .$$ The basic properties of density functions are given by the following claim. Claim 3.3.16 (Properties of continuous random variables) If $f_X$ is a density function then i. $f_X(x) \geq 0$ for all $x \in \mathbb{R}$. ii. $\int{-\infty}^{\infty} f_X(x) d x=1$.
The first property is a direct consequence of Definition $3.3 .14$ and the second property can be seen directly from part ii. of Proposition 3.2.3.We use the same notation (lowercase $f$ ) for density functions as we do for mass functions. This serves to emphasise that the density plays the same role for a continuous variable as mass does for a discrete variable. However, there is an important distinction; it is legitimate to have density function values that are greater than one since values of a density function do not give probabilities. Probability is not associated with the values of the density function but with the area beneath the curve that the density function defines. In order to work out the probability that a random variable $X$ takes a value between $a$ and $b$, we work out the area above the $x$-axis, beneath the density, and between the lines $x=a$ and $x=b$. As we might expect, given this interpretation, the total area beneath a density function is one, as stated in Claim 3.3.16. The general relationship between probability and density is given by the following proposition.

## 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.

# 统计推断代考

## 统计代写|统计推断代写统计推理代考|连续随机变量和密度函数

.

$$F_X(x)=\int_{-\infty}^x f_X(u) d u \quad \text { for } x \in \mathbb{R}$$
。函数$f_X$被称为$X$的(概率)密度函数

$$f_X(x)=\left.\frac{d}{d u} F_X(u)\right|{u=x}=F_X^{\prime}(x) \text { for all } x \in \mathbb{R} .$$密度函数的基本性质由以下权利要求给出。权利要求3.3.16(连续随机变量的属性)如果$f_X$是一个密度函数，那么i. $f_X(x) \geq 0$对所有$x \in \mathbb{R}$。2$\int{-\infty}^{\infty} f_X(x) d x=1$ .

## 统计代写|统计推断代写统计推断代考|分布的参数和族

.

1. 显示出来 $\Gamma(1 / 2)=\sqrt{\pi}$，从而写出一个公式 $\Gamma(k / 2)$，其中 $k$ 为正整数。[提示:在的定义中。 $\Gamma(1 / 2)$，使用代换法 $z=\sqrt{2 u}$ 重新排列被积函数得到标准的法向密度函数。
2. (几何质量函数)表示函数
$$f_X(x)=(1-p)^{x-1} p \text { for } x=1,2, \ldots,$$
是有效的质量函数。
3. (几何累积分布函数)设 $X \sim \operatorname{Geometric}(p)$。的累积分布函数 $X$.
4. (负二项式质量函数)表示函数
$$f_X(x)=\left(\begin{array}{l} x-1 \ r-1 \end{array}\right) p^r(1-p)^{x-r} \text { for } x=r, r+1, \ldots$$
是有效的质量函数。
5. (连续均匀的累积分布函数)设 $X \sim$ Unif $[a, b]$。的累积分布函数 $X$.
6. 显示函数
$$f_X(x)= \begin{cases}\frac{3}{2} x^2+x & \text { for } 0 \leq x \leq 1 \ 0 & \text { otherwise }\end{cases}$$
是有效的密度函数。

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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。

assignmentutor™您的专属作业导师
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