## 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.
4.4.1 Independence for pairs of random variables
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
$$F_{X, Y}(x, y)=F_X(x) F_Y(y) \quad \text { for all } x, y \in \mathbb{R}$$
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.

## 统计代写|统计推断代写Statistical inference代考|Transformations of continuous random variables

Recall from subsection $3.6 .2$ that if $Y$ is given by a strictly monotone function of a continuous random variable $X$, we can derive an expression for the density of $Y$ in terms of the density of $X$. We extend these ideas to functions of several continuous random variables, starting as usual with the bivariate case. We require a standard definition first.
Definition 4.6.1 (One-to-one and onto) Consider a function $g$ with domain $D$ and range $R$. We say that $g$ is:
i. one-to-one if $g\left(x_1\right)=g\left(x_2\right) \Rightarrow x_1=x_2$ for all $x_1, x_2 \in D$,
ii. onto if for all $y \in R$ we can find $x \in D$ such that $y=g(x)$.
4.6.1 Bivariate transformations
We are interested in transforming one pair of random variables into another pair of random variables. Consider pairs of random variables $(U, V)$ and $(X, Y)$. Suppose that $X$ and $Y$ are both functions of $U$ and $V$, say
\begin{aligned} &X=g_1(U, V), \ &Y=g_2(U, V) . \end{aligned}
We will make extensive use of the inverse transformation,
\begin{aligned} &U=h_1(X, Y) \ &V=h_2(X, Y) . \end{aligned}
We label the overall transformation as $g$, so $(X, Y)=g(U, V)$, and the inverse as $\boldsymbol{h}$, so $(U, V)=g^{-1}(X, Y)=\boldsymbol{h}(X, Y)$. Suppose $g$ is a well-behaved function $g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$.

# 统计推断代考

## 统计代写|统计推断代写统计推断代考|独立随机变量

4.4.1随机变量对的独立性

$$F_{X, Y}(x, y)=F_X(x) F_Y(y) \quad \text { for all } x, y \in \mathbb{R}$$
，根据定义4.4.1，事件${X \leq x}$和${Y \leq y}$是独立的，当且仅当它们相交的概率是单个概率的乘积。权利要求4.4.2指出，对于独立随机变量，对边际的了解等价于对联合分布的了解;这是一个吸引人的性质。这一主张可以用质量或密度来重申

## 统计代写|统计推断代写统计推断代考|连续随机变量的变换

i。一对一的if $g\left(x_1\right)=g\left(x_2\right) \Rightarrow x_1=x_2$ 为所有人 $x_1, x_2 \in D$，
ii。on if for all $y \in R$ 我们可以找到 $x \in D$ 如此这般 $y=g(x)$我们感兴趣的是将一对随机变量转换为另一对随机变量。考虑成对的随机变量 $(U, V)$ 和 $(X, Y)$。假设 $X$ 和 $Y$ 都是 $U$ 和 $V$，输入
\begin{aligned} &X=g_1(U, V), \ &Y=g_2(U, V) . \end{aligned}我们将广泛地使用逆变换，
\begin{aligned} &U=h_1(X, Y) \ &V=h_2(X, Y) . \end{aligned}我们将整个转换标记为 $g$，所以 $(X, Y)=g(U, V)$，其逆为 $\boldsymbol{h}$，所以 $(U, V)=g^{-1}(X, Y)=\boldsymbol{h}(X, Y)$。假设 $g$ 是一个行为良好的函数吗 $g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$.

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