### 统计代写|统计推断代写Statistical inference代考|STATS 2107

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• 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|Covariance and correlation

In the univariate case, we discussed the use of single-number summaries for the features of a distribution. For example, we might use the mean as a measure of central tendency and the variance as a measure of spread. In the multivariate case we might, in addition, be interested in summarising the dependence between random variables. For a pair of random variables, a commonly used quantity for measuring the degree of (linear) association is correlation. The starting point for the definition of correlation is the notion of covariance.
Definition 4.3.5 (Covariance)
For random variables $X$ and $Y$, the covariance between $X$ and $Y$ is defined as
$$\operatorname{Cov}(X, Y)=\mathbb{E}[(X-\mathbb{E}(X))(Y-\mathbb{E}(Y))]$$
An alternative form for the covariance is
$$\operatorname{Cov}(X, Y)=\mathbb{E}(X Y)-\mathbb{E}(X) \mathbb{E}(Y)$$
Proving the equivalence of these two forms is part of Exercise 4.3. Covariance has a number of properties that are immediate consequences of its definition as an expectation.
Claim 4.3.6 (Properties of covariance)
For random variables $X, Y, U$, and $V$ the covariance has the following properties.
i. $\operatorname{Symmetry:} \operatorname{Cov}(X, Y)=\operatorname{Cov}(Y, X)$.

## 统计代写|统计推断代写Statistical inference代考|Joint moments

Joint moments provide information about the dependence structure between two random variables. For most practical purposes we only consider joint moments of low order. An example of such a joint moment is the covariance.
Definition 4.3.10 (Joint moments and joint central moments) If $X$ and $Y$ are random variables, then the $(r, s)^{\text {th }}$ joint moment of $X$ and $Y$ is
$$\mu_{r, s}^{\prime}=\mathbb{E}\left(X^{r} Y^{s}\right)$$
The $(r, s)^{\text {th }}$ joint central moment of $X$ and $Y$ is
$$\mu_{r, s}=\mathbb{B}\left[(X-\mathbb{E}(X))^{r}(Y-\mathbb{E}(Y))^{s}\right]$$
Many familiar quantities can be expressed as joint moments.
i. $r^{\text {th }}$ moment for $X: \mathbb{E}\left(X^{r}\right)=\mu_{r, 0^{\prime}}^{\prime}$.
ii. $r^{\text {th }}$ central moment for $X: \mathbb{E}\left[\left(X-\mu_{X}\right)^{r}\right]=\mu_{r, 0}$.
iii. Covariance: $\operatorname{Cov}(X, Y)=\mu_{1,1}$.
iv. Correlation: $\operatorname{Corr}(X, Y)=\mu_{1,1} / \sqrt{\mu_{2,0} \mu_{0,2}}$.
Joint moments are evaluated using Proposition 4.3.1. We return to the simple polynomial density of Example $4.2 .10$ to illustrate.

## 统计代写|统计推断代写Statistical inference代考|Joint moment-generating functions

The joint moments of a distribution are encapsulated in the joint moment-generating function.
Definition 4.3.12 (Joint moment-generating function)
For random variables $X$ and $Y$, the joint moment-generating function is defined as
$$M_{X, Y}(t, u)=\mathrm{E}\left(e^{t X+u Y}\right)$$
The argument of the expectation operator in the definition of the joint momentgenerating function can be written as the product of two series. Assuming we can swap the order of summations and expectations, we have
$$M_{X, Y}(t, u)=\mathbb{B}\left(e^{t X} e^{u Y}\right)=\mathbb{B}\left(\sum_{i=0}^{\infty} \frac{(t X)^{i}}{i !} \sum_{j=0}^{\infty} \frac{(u Y)^{j}}{j !}\right)=\sum_{i=0}^{\infty} \sum_{j=0}^{\infty} \mathbb{B}\left(X^{i} Y^{j}\right) \frac{t^{i} u^{j}}{i ! j !}$$
Thus, the joint moment-generating function is a polynomial in $t$ and $u$. The $(r, s)^{\text {th }}$ joint moment is the coefficient of $\left(t^{r} u^{s}\right) /(r ! s !)$ in the polynomial expansion of the joint moment-generating function. One consequence is that the joint moments can be evaluated by differentiation:
$$M_{X, Y}^{(r, s)}(0,0)=\left.\frac{d^{r+s}}{d t^{r} d u^{s}} M_{X, Y}(t, u)\right|{t=0, u=0}=\mathbb{B}\left(X^{r} Y^{s}\right)=\mu{r, s^{-}}^{\prime}$$
The marginal moment-generating functions can be recovered from the joint,
\begin{aligned} &M_{X}(t)=\mathbb{B}\left(e^{t X}\right)=M_{X, Y}(t, 0) \ &M_{Y}(t)=\mathbb{B}\left(e^{u Y}\right)=M_{X, Y}(0, u) \end{aligned}

## 统计代写|统计推断代写Statistical inference代考|Joint moments

μr,s′=和(Xr是s)

μr,s=乙[(X−和(X))r(是−和(是))s]

ii.rth 中心时刻X:和[(X−μX)r]=μr,0.
iii. 协方差：这⁡(X,是)=μ1,1.
iv. 相关性：更正⁡(X,是)=μ1,1/μ2,0μ0,2.

## 有限元方法代写

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## MATLAB代写

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