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# 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Distribution and Density Function

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## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Distribution and Density Function

Let $X=\left(X_1, X_2, \ldots, X_p\right)^{\top}$ be a random vector. The cumulative distribution function (cdf) of $X$ is defined by
$$F(x)=P(X \leq x)=P\left(X_1 \leq x_1, X_2 \leq x_2, \ldots, X_p \leq x_p\right)$$
For continuous $X$, there exists a nonnegative probability density function (pdf) $f$, such that
$$F(x)=\int_{-\infty}^x f(u) d u$$
Note that
$$\int_{-\infty}^{\infty} f(u) d u=1$$

Most of the integrals appearing below are multidimensional. For instance, $\int_{-\infty}^x f(u) d u$ means $\int_{-\infty}^{x_p} \cdots \int_{-\infty}^{x_1} f\left(u_1, \ldots, u_p\right) d u_1 \cdots d u_p$. Note also that the cdf $F$ is differentiable with
$$f(x)=\frac{\partial^p F(x)}{\partial x_1 \cdots \partial x_p} .$$
For discrete $X$, the values of this random variable are concentrated on a countable or finite set of points $\left{c_j\right}_{j \in J}$, the probability of events of the form ${X \in D}$ can then be computed as
$$P(X \in D)=\sum_{\left{j: c_j \in D\right}} P\left(X=c_j\right) .$$
If we partition $X$ as $X=\left(X_1, X_2\right)^{\top}$ with $X_1 \in \mathbb{R}^k$ and $X_2 \in \mathbb{R}^{p-k}$, then the function
$$F_{X_1}\left(x_1\right)=P\left(X_1 \leq x_1\right)=F\left(x_{11}, \ldots, x_{1 k}, \infty, \ldots, \infty\right)$$
is called the marginal cdf. $F=F(x)$ is called the joint cdf. For continuous $X$ the marginal pdf can be computed from the joint density by “integrating out” the variable not of interest.
$$f_{X_1}\left(x_1\right)=\int_{-\infty}^{\infty} f\left(x_1, x_2\right) d x_2 .$$
The conditional pdf of $X_2$ given $X_1=x_1$ is given as
$$f\left(x_2 \mid x_1\right)=\frac{f\left(x_1, x_2\right)}{f_{X_1}\left(x_1\right)}$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Moments—Expectation and Covariance Matrix

If $X$ is a random vector with density $f(x)$ then the expectation of $X$ is
$$E X=\left(\begin{array}{c} E X_1 \ \vdots \ E X_p \end{array}\right)=\int x f(x) d x=\left(\begin{array}{c} \int x_1 f(x) d x \ \vdots \ \int x_p f(x) d x \end{array}\right)=\mu .$$
Accordingly, the expectation of a matrix of random elements has to be understood component by component. The operation of forming expectations is linear:
$$E(\alpha X+\beta Y)=\alpha E X+\beta E Y$$
If $\mathcal{A}(q \times p)$ is a matrix of real numbers, we have:
$$E(\mathcal{A} X)=\mathcal{A} E X$$
When $X$ and $Y$ are independent,
$$E\left(X Y^{\top}\right)=E X E Y^{\top}$$
The matrix
$$\operatorname{Var}(X)=\Sigma=E(X-\mu)(X-\mu)^{\top}$$
is the (theoretical) covariance matrix. We write for a vector $X$ with mean vector $\mu$ and covariance matrix $\Sigma$,
$$X \sim(\mu, \Sigma)$$
The $(p \times q)$ matrix
$$\Sigma_{X Y}=\operatorname{Cov}(X, Y)=E(X-\mu)(Y-\nu)^{\top}$$
is the covariance matrix of $X \sim\left(\mu, \Sigma_{X X}\right)$ and $Y \sim\left(\nu, \Sigma_{Y Y}\right)$. Note that $\Sigma_{X Y}=\Sigma_{Y X}^{\top}$ and that $Z=\left(\begin{array}{l}X \ Y\end{array}\right)$ has covariance $\Sigma_{Z Z}=\left(\begin{array}{c}\Sigma_{X X} \Sigma_{X Y} \ \Sigma_{Y X} \Sigma_{Y Y}\end{array}\right)$. From
$$\operatorname{Cov}(X, Y)=E\left(X Y^{\top}\right)-\mu \nu^{\top}=E\left(X Y^{\top}\right)-E X E Y^{\top}$$
it follows that $\operatorname{Cov}(X, Y)=0$ in the case where $X$ and $Y$ are independent. We often say that $\mu=E(X)$ is the first order moment of $X$ and that $E\left(X X^{\top}\right)$ provides the second order moments of $X$ :
$$E\left(X X^{\top}\right)=\left{E\left(X_i X_j\right)\right}, \text { for } i=1, \ldots, p \text { and } j=1, \ldots, p \text {. }$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Distribution and Density Function

$$F(x)=P(X \leq x)=P\left(X_1 \leq x_1, X_2 \leq x_2, \ldots, X_p \leq x_p\right)$$

$$F(x)=\int_{-\infty}^x f(u) d u$$

$$\int_{-\infty}^{\infty} f(u) d u=1$$

$$f(x)=\frac{\partial^p F(x)}{\partial x_1 \cdots \partial x_p} .$$

$$F_{X_1}\left(x_1\right)=P\left(X_1 \leq x_1\right)=F\left(x_{11}, \ldots, x_{1 k}, \infty, \ldots, \infty\right)$$

$$f_{X_1}\left(x_1\right)=\int_{-\infty}^{\infty} f\left(x_1, x_2\right) d x_2 .$$

$$f\left(x_2 \mid x_1\right)=\frac{f\left(x_1, x_2\right)}{f_{X_1}\left(x_1\right)}$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Moments—Expectation and Covariance Matrix

$$E X=\left(\begin{array}{c} E X_1 \ \vdots \ E X_p \end{array}\right)=\int x f(x) d x=\left(\begin{array}{c} \int x_1 f(x) d x \ \vdots \ \int x_p f(x) d x \end{array}\right)=\mu .$$

$$E(\alpha X+\beta Y)=\alpha E X+\beta E Y$$

$$E(\mathcal{A} X)=\mathcal{A} E X$$

$$E\left(X Y^{\top}\right)=E X E Y^{\top}$$

$$\operatorname{Var}(X)=\Sigma=E(X-\mu)(X-\mu)^{\top}$$

$$X \sim(\mu, \Sigma)$$
$(p \times q)$矩阵
$$\Sigma_{X Y}=\operatorname{Cov}(X, Y)=E(X-\mu)(Y-\nu)^{\top}$$

$$\operatorname{Cov}(X, Y)=E\left(X Y^{\top}\right)-\mu \nu^{\top}=E\left(X Y^{\top}\right)-E X E Y^{\top}$$

$$E\left(X X^{\top}\right)=\left{E\left(X_i X_j\right)\right}, \text { for } i=1, \ldots, p \text { and } j=1, \ldots, p \text {. }$$

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