## 统计代写|统计推断代写Statistical inference代考|Joint Moments of Random Variables

$\Lambda \mathrm{s}$ in the case of univariate distributions, the best way to interpret the unknown parameters is via the moments. In direct analogy to the univariate case, we define the joint product moments of $\operatorname{order}(k, m)$ by
$$\mu_{k m}^{\prime}=E\left{X^k Y^m\right}, k, m=0,1,2, \ldots$$
and the joint central moments of order $(k, m)$ by
$$\mu_{k m}=E\left{(X-E(X))^k(Y-E(Y))^m\right}, k, m=0,1,2, \ldots$$
The first two joint product and central moments are
$$\begin{array}{ll} \mu_{10}^{\prime}=E(X), \mu_{01}^{\prime}=E(Y), & \mu_{10}=0, \mu_{01}=0 \ \mu_{20}^{\prime}=E(X)^2+\operatorname{Var}(X), & \mu_{20}=\operatorname{Var}(X) \ \mu_{02}^{\prime}=E(Y)^2+\operatorname{Var}(Y), & \mu_{02}=\operatorname{Var}(Y), \ \mu_{11}^{\prime}=E(X \cdot Y), & \mu_{11}=E[(X-E(X))(Y-E(Y))] . \end{array}$$
The most important and widely used joint moment is the covariance, defined by
$$\mu_{11}:=\operatorname{Cov}(X, Y)=E{[X-E(X)][Y-E(Y)]} .$$
Example 4.6 Consider the joint Normal distribution whose density is given in (4.7). We know from Chapter 3 that the parameters $\left(\mu_1, \mu_2, \sigma_{11}, \sigma_{22}\right)$ correspond to the moments:
$$\mu_1=E(Y), \quad \mu_2=E(X), \quad \sigma_{11}=\operatorname{Var}(Y), \quad \sigma_{22}=\operatorname{Var}(X) .$$
The additional parameter $\sigma_{12}$ turns out to be the covariance between the two random variables, i.e. $\sigma_{12}:=\operatorname{Cov}(X, Y)$.

## 统计代写|统计推断代写Statistical inference代考|The n Random Variables Joint Distribution

Extending the concept of a random variable from a two-dimensional to an $n$-dimensional random vector $\mathbf{X}(.):=\left(X_1(.), X_2(.), \ldots, X_n(.)\right)$ is straightforward:
$$\mathbf{X}(.): S \rightarrow \mathbb{R}^n,$$
where $\mathbb{R}^n:=\mathbb{R} \times \mathbb{R} \times \cdots \times \mathbb{R}$ denotes the Cartesian product of the real line (Chapter 2 ). The $n$-variable function $\mathbf{X}($.) is said to be a random vector relative to $\Im$ if
$\mathbf{X}(.): S \rightarrow \mathbb{R}^n$, such that $\mathbf{X}^{-1}((-\infty, \mathbf{x}]) \in \mathfrak{3}$, for all $\mathbf{x} \in \mathbb{R}_X^n$,
where $\mathbf{x}:=\left(x_1, x_2, \ldots, x_n\right)$ and $(-\infty, \mathbf{x}]:=\left(-\infty, x_1\right] \times\left(-\infty, x_2\right] \times \cdots \times\left(-\infty, x_n\right]$. Note that all the random variables $\left(X_1(.), X_2(.), \ldots, X_n(.)\right)$ are defined on the same outcomes set $S$ and relative to the same event space $\Re$.

In view of the fact that $\Im$ is a $\sigma$-field, we know that $\mathbf{X}:=\left(X_1, X_2, \ldots, X_n\right)$ is a random vector relative to $\Im$ if and only if the random variables $\left(X_1, X_2, \ldots, X_n\right)$ are random variables relative to $\Im$. This is because $X_k^{-1}\left(-\infty, x_k\right] \in \Im$ for all $k=1,2, \ldots, n$, and so does their intersection:
$$\left(\bigcap_{k=1}^n X_k^{-1}\left(-\infty, x_k\right]\right) \in \mathcal{\aleph} .$$
The concepts introduced above for the two random variable case can easily be extended to the $n$ random variable case, satisfying similar properties as shown in Table 4.4.

## 统计代写|统计推断代写统计推断代考|随机变量关节矩

$\Lambda \mathrm{s}$在单变量分布的情况下，解释未知参数的最佳方式是通过矩。与单变量情况直接类似，我们用
$$\mu_{k m}^{\prime}=E\left{X^k Y^m\right}, k, m=0,1,2, \ldots$$

$$\mu_{k m}=E\left{(X-E(X))^k(Y-E(Y))^m\right}, k, m=0,1,2, \ldots$$

$$\begin{array}{ll} \mu_{10}^{\prime}=E(X), \mu_{01}^{\prime}=E(Y), & \mu_{10}=0, \mu_{01}=0 \ \mu_{20}^{\prime}=E(X)^2+\operatorname{Var}(X), & \mu_{20}=\operatorname{Var}(X) \ \mu_{02}^{\prime}=E(Y)^2+\operatorname{Var}(Y), & \mu_{02}=\operatorname{Var}(Y), \ \mu_{11}^{\prime}=E(X \cdot Y), & \mu_{11}=E[(X-E(X))(Y-E(Y))] . \end{array}$$

$$\mu_{11}:=\operatorname{Cov}(X, Y)=E{[X-E(X)][Y-E(Y)]} .$$

$$\mu_1=E(Y), \quad \mu_2=E(X), \quad \sigma_{11}=\operatorname{Var}(Y), \quad \sigma_{22}=\operatorname{Var}(X) .$$

## 统计代写|统计推断代写统计推断代考| n随机变量联合分布

$$\mathbf{X}(.): S \rightarrow \mathbb{R}^n,$$

$\mathbf{X}(.): S \rightarrow \mathbb{R}^n$，则$\mathbf{X}^{-1}((-\infty, \mathbf{x}]) \in \mathfrak{3}$对于所有$\mathbf{x} \in \mathbb{R}_X^n$，

$$\left(\bigcap_{k=1}^n X_k^{-1}\left(-\infty, x_k\right]\right) \in \mathcal{\aleph} .$$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: