# 统计代写|统计推断代写Statistical inference代考|STAT3923

#### Doug I. Jones

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
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## 统计代写|统计推断代写Statistical inference代考|Expectation, variance, and higher moments

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 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.

Definition 3.4.1 (Mean)
The mean of a random variable $X$, denoted $\mathbb{E}(X)$, is given by
$$\mathbb{E}(X)= \begin{cases}\sum_x x f_X(x) & \text { if } X \text { discrete, } \ \int_{-\infty}^{\infty} x f_X(x) d x & \text { if } X \text { continuous, }\end{cases}$$
where, to guarantee that $\mathbb{B}(X)$ is well defined, we usually insist that $\Sigma_x|x| f_X(x)<\infty$ in the discrete case and $\int_{-\infty}^{\infty}|x| f_X(x) d x<\infty$ in the continuous case.

The Greek letter $\mu$ is often used to denote the mean. In this context it is clear that the mean may be viewed as a parameter of the distribution. The mean of $X$ is often also referred to as the expected value of $X$ or the expectation of $X$. During our discussion of the properties of random variables we will also use $\mathrm{B}(X)$ to denote the mean. The expectation operator $\mathbb{E}($.$) defines a process of averaging; this process is$ widely applied and is the subject of the next subsection.

When talking about expectation, we will not repeatedly say “whenever this expectation exists”. In many propositions about the mean, there is an implicit assumption that the proposition only holds when the mean is well defined. The same applies in later sections to other quantities defined as an expectation, such as the variance, higher moments, and moment-generating functions (more on these later).

We give two examples of computation of the mean using Definition 3.4.1, one a discrete random variable and the other continuous.

## 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{E}\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, $(X-\mathbb{E}(X))^2$ is a positive random variable, so $\operatorname{Var}(X)=\mathbb{E}[(X-$ $\left.\mathbb{E}(X))^2\right] \geq 0$ 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 $Y-\mathbb{E}(Y)=a_1(X-\mathbb{E}(X))$ and so
\begin{aligned} \operatorname{Var}\left(a_0+a_1 X\right) &=\operatorname{Var}(Y)=\mathbb{E}\left[(Y-\mathbb{E}(Y))^2\right]=\mathbb{E}\left[a_1^2(X-\mathbb{E}(X))^2\right] \ &=a_1^2 \operatorname{Var}(X) \end{aligned}

# 统计推断代考

## 统计代写|统计推断代写统计推断代考|期望、方差和更高矩

3.4.1(均值)

$$\mathbb{E}(X)= \begin{cases}\sum_x x f_X(x) & \text { if } X \text { discrete, } \ \int_{-\infty}^{\infty} x f_X(x) d x & \text { if } X \text { continuous, }\end{cases}$$

## 统计代写|统计推断代写统计推断代考|随机变量方差

\begin{aligned} \sigma^2 &=\operatorname{Var}(X)=\mathbb{E}\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}
，只要这个和/积分是有限的。标准差定义为$\sigma=$$\sqrt{\operatorname{Var}(X)} 方差算符的一些性质由以下命题给出。命题3.4.8(方差的性质) 对于随机变量X和实常数a_0和a_1，方差具有以下性质: i。\operatorname{Var}(X) \geq 0， ii。\operatorname{Var}\left(a_0+a_1 X\right)=a_1^2 \operatorname{Var}(X) . 证明。 这两个属性都继承自方差作为期望的定义。根据定义，(X-\mathbb{E}(X))^2是一个正随机变量，因此，根据Claim 3.4.6, \operatorname{Var}(X)=\mathbb{E}[(X-$$\left.\mathbb{E}(X))^2\right] \geq 0$ .
ii。如果我们根据期望的线性度定义$Y=a_0+a_1 X$，那么$\mathbb{E}(Y)=a_0+a_1 \mathbb{E}(X)$。因此$Y-\mathbb{E}(Y)=a_1(X-\mathbb{E}(X))$ and so
\begin{aligned} \operatorname{Var}\left(a_0+a_1 X\right) &=\operatorname{Var}(Y)=\mathbb{E}\left[(Y-\mathbb{E}(Y))^2\right]=\mathbb{E}\left[a_1^2(X-\mathbb{E}(X))^2\right] \ &=a_1^2 \operatorname{Var}(X) \end{aligned}

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