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

## 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$\mathbf{x} \in \mathbb{R}_X^n$，

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

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