### 统计代写|统计推断代写Statistical inference代考|STAT 7604

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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|Independent random variables

The term independent and identically distributed (IID) is one that is used with great frequency in statistics. One of the key assumptions that is often made in inference is that we have a random sample. Assuming a sample is random is equivalent to stating that a reasonable model for the process that generates the data is a sequence of independent and identically distributed random variables. We start by defining what it means for a pair of random variables to be independent.

Definition 4.4.1 (Independent random variables)
The random variables $X$ and $Y$ are independent if and only if the events ${X \leq x}$ and ${Y \leq y}$ are independent for all $x$ and $y$.

One immediate consequence of this definition is that, for independent random variables, it is possible to generate the joint distribution from the marginal distributions.
Claim 4.4.2 (Joint distribution of independent random variables)
Random variables $X$ and $Y$ are independent if and only if the joint cumulative distribution function of $X$ and $Y$ is the product of the marginal cumulative distribution functions, that is, if and only if
$$F_{X, Y}(x, y)=F_{X}(x) F_{Y}(y) \text { for all } x, y \in \mathbb{R}$$
The claim holds since, by Definition 4.4.1, the events ${X \leq x}$ and ${Y \leq y}$ are independent if and only if the probability of their intersection is the product of the individual probabilities. Claim 4.4.2 states that, for independent random variables, knowledge of the margins is equivalent to knowledge of the joint distribution; this is an attractive property. The claim can be restated in terms of mass or density.
Proposition 4.4.3 (Mass/density of independent random variables)
The random variables $X$ and $Y$ are independent if and only if their joint mass/density is the product of the marginal mass/density functions, that is, if and only if
$$f_{X, Y}(x, y)=f_{X}(x) f_{Y}(y) \quad \text { for all } x, y \in \mathbb{R}$$
Proof.

## 统计代写|统计推断代写Statistical inference代考|Mutual independence

We can readily extend the ideas of this section to a sequence of $n$ random variables. When considering many random variables, the terms pairwise independent and mutually independent are sometimes used. Pairwise independent, as the name suggests, means that every pair is independent in the sense of Definition 4.4.1.

Definition 4.4.7 (Mutually independent random variables)
The random variables $X_{1}, X_{2}, \ldots, X_{n}$ are mutually independent if and only if the events $\left{X_{1} \leq x_{1}\right},\left{X_{2} \leq x_{2}\right}, \ldots,\left{X_{n} \leq x_{n}\right}$ are mutually independent for all choices of $x_{1}, x_{2}, \ldots, x_{n}$

When $X_{1}, X_{2}, \ldots, X_{n}$ are mutually independent the term “mutually” is often dropped and we just say $X_{1}, X_{2}, \ldots, X_{n}$ are independent or $\left{X_{i}\right}$ is a sequence of independent random variables. Note that this is a stronger property than pairwise independence; mutually independent implies pairwise independent but the reverse implication does not hold.

Any one of the equivalent statements summarised in the following claim could be taken to be a definition of independence.
Claim 4.4.8 (Equivalent statements of mutual independence) If $X_{1}, \ldots, X_{n}$ are random variables, the following statements are equivalent:
i. The events $\left{X_{1} \leq x_{1}\right},\left{X_{2} \leq x_{2}\right}, \ldots,\left{X_{n} \leq x_{n}\right}$ are independent for all $x_{1}, \ldots, x_{n}$.
ii. $F_{X_{1}, \ldots, X_{n}}\left(x_{1}, \ldots, x_{n}\right)=F_{X_{1}}\left(x_{1}\right) F_{X_{2}}\left(x_{2}\right) \ldots F_{X_{n}}\left(x_{n}\right)$ for all $x_{1}, \ldots, x_{n}$.
iii. $f_{X_{1}, \ldots, X_{n}}\left(x_{1}, \ldots, x_{n}\right)=f_{X_{1}}\left(x_{1}\right) f_{X_{2}}\left(x_{2}\right) \ldots f_{X_{n}}\left(x_{n}\right)$ for all $x_{1}, \ldots, x_{n}$.
The implications of mutual independence may be summarised as follows.
Claim 4.4.9 (Implications of mutual independence)
If $X_{1}, \ldots, X_{n}$ are mutually independent random variables, then
i. $\mathrm{E}\left(X_{1} X_{2} \ldots X_{n}\right)=\mathrm{E}\left(X_{1}\right) \mathrm{E}\left(X_{2}\right) \ldots \mathrm{E}\left(X_{n}\right)$,
ii. if, in addition, $g_{1}, \ldots, g_{n}$ are well-behaved, real-valued functions, then the random variables $g_{1}\left(X_{1}\right), \ldots, g_{n}\left(X_{n}\right)$ are also mutually independent.

## 统计代写|统计推断代写Statistical inference代考|Identical distributions

Another useful simplifying assumption is that of identical distributions.
Definition 4.4.10 (Identically distributed random variables)
The random variables $X_{1}, X_{2}, \ldots, X_{n}$ are identically distributed if and only if their cumulative distribution functions are identical, that is
$$F_{X_{1}}(x)=F_{X_{2}}(x)=\ldots=F_{X_{n}}(x) \text { for all } x \in \mathbb{R}$$
If $X_{1}, X_{2}, \ldots, X_{n}$ are identically distributed we will often just use the letter $X$ to denote a random variable that has the distribution common to all of them. So the cumulative distribution function of $X$ is $\mathrm{P}(X \leq x)=F_{X}(x)=F_{X_{1}}(x)=\ldots=F_{X_{n}}(x)$. If $X_{1}, X_{2}, \ldots, X_{n}$ are independent and identically distributed, we may sometimes denote this as $\left{X_{i}\right} \sim$ IID.

1. Suppose $X_{1}, \ldots, X_{n}$ is a sequence of $n$ independent and identically distributed standard normal random variables. Find an expression for the joint density of $X_{1}, \ldots, X_{n}$. [We denote this by $\left{X_{i}\right} \sim \operatorname{NID}(0,1)$, where NID stands for “normal and independently distributed”.]
2. Let $X_{1}, \ldots, X_{n}$ be a sequence of $n$ independent random variables with cumulantgenerating functions $K_{X_{1}}, \ldots, K_{X_{n}}$. Find an expression for the joint cumulantgenerating function $K_{X_{1}, \ldots, X_{n}}$ in terms of the individual cumulant-generating functions.

## 统计代写|统计推断代写Statistical inference代考|Independent random variables

FX,是(X,是)=FX(X)F是(是) 对所有人 X,是∈R

FX,是(X,是)=FX(X)F是(是) 对所有人 X,是∈R

## 统计代写|统计推断代写Statistical inference代考|Mutual independence

i．事件\left{X_{1} \leq x_{1}\right},\left{X_{2} \leq x_{2}\right}, \ldots,\left{X_{n} \leq x_{n} \正确的}\left{X_{1} \leq x_{1}\right},\left{X_{2} \leq x_{2}\right}, \ldots,\left{X_{n} \leq x_{n} \正确的}对所有人都是独立的X1,…,Xn.
ii.FX1,…,Xn(X1,…,Xn)=FX1(X1)FX2(X2)…FXn(Xn)对所有人X1,…,Xn.
iii.FX1,…,Xn(X1,…,Xn)=FX1(X1)FX2(X2)…FXn(Xn)对所有人X1,…,Xn.

i.和(X1X2…Xn)=和(X1)和(X2)…和(Xn),

## 统计代写|统计推断代写Statistical inference代考|Identical distributions

FX1(X)=FX2(X)=…=FXn(X) 对所有人 X∈R

1. 认为X1,…,Xn是一个序列n独立同分布的标准正态随机变量。求联合密度的表达式X1,…,Xn. [我们将其表示为\left{X_{i}\right} \sim \operatorname{NID}(0,1)\left{X_{i}\right} \sim \operatorname{NID}(0,1)，其中 NID 代表“正常且独立分布”。]
2. 让X1,…,Xn是一个序列n具有累积量生成函数的独立随机变量ķX1,…,ķXn. 找到联合累积量生成函数的表达式ķX1,…,Xn就各个累积量生成函数而言。

## 有限元方法代写

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## MATLAB代写

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