## 统计代写|统计推断代写Statistical inference代考|A more thorough treatment of random variables

In earlier sections of this chapter we refer rather vaguely to conditions on a set $B$ for $\mathrm{P}(X \in B)$ to be well defined and conditions on a function $g$ for $g(X)$ to be a random variable. We also suggest that we are not really interested in random variables as maps and that, for many situations, the notion of an underlying sample space is not particularly useful. In this section, we attempt to provide some justification for these assertions. The material here is technical and may be excluded without affecting understanding of other parts of the text. We start by providing an alternative definition of a random variable. This is equivalent to Definition 3.1.2 but uses more abstract concepts; key among them is the Borel $\sigma$-algebra.
Definition 3.8.1 (Borel $\sigma$-algebra on $\mathbb{R}$ )
Let $C$ be the collection of all open intervals of $\mathbb{R}$. The Borel $\sigma$-algebra on $\mathbb{R}$ is the (unique) smallest $\sigma$-algebra that contains $C$. We denote the Borel $\sigma$-algebra on $\mathbb{R}$ by $\mathcal{B}$. An element of $\mathcal{B}$ is referred to as a Borel set.

From the definition it is clear that any open interval $(x, y)$ is a Borel set. It is also the case that closed intervals $[x, y]$. half-open intervals $(x, y]$ and $[x, y)$, and finite unions of interval are all Borel sets in $\mathbb{R}$. In fact, sets that are not Borel sets are hard to construct; any subset of $\mathbb{R}$ that you come across in a practical problem is likely to be a Borel set. Clearly, since $\mathcal{B}$ is a $\sigma$-algebra, $(\mathbb{R}, \mathcal{B})$ is a measurable space. The term measurable can also be applied to functions.
Definition 3.8.2 (Measurable function)
Consider measurable spaces $(\Omega, \mathcal{F})$ and $(\mathbb{R}, \mathcal{B})$. We say that a function $h: \Omega \rightarrow \mathbb{R}$ is $\mathcal{F}$-measurable if $h^{-1}(R) \in \mathcal{F}$ for all $B \in \mathcal{B}$.
We can now give an alternative definition of a random variable.

Definition 3.8.3 (Random variable)
For a probability space $(\Omega, \mathcal{F}, \mathrm{P})$ and measurable space $(\mathbb{R}, \mathcal{B})$, a random variable, $X$, is a measurable function $X: \Omega \rightarrow \mathbb{R}$.

For every random variable. $X$, we can define a measure on $\mathbb{R}$ that completely characterises the distribution of probability associated with $X$. This measure is sometimes referred to as the law of $X$; a non-rigorous account follows. Suppose that we have a probability space $(\Omega, \mathcal{F}, \mathrm{P})$ and that $B$ is a Borel set, that is, $B \subseteq \mathbb{R}$ and $B \in \mathcal{B}$. Our definition of a random variable ensures that the inverse image of $B$ under $X$ is an event, that is, $X^{-1}(B) \in \mathcal{F}$. As such, it makes sense to talk about the probability of the inverse image,
$$\mathrm{P}\left(X^{-1}(B)\right)=\mathrm{P}({\omega \in \Omega: X(\omega) \in B}=\mathrm{P}(X \in B) .$$
If we define $Q_X(B)=\mathrm{P}(X \in B)$, then $Q_X: \mathcal{B} \rightarrow[0,1]$ and $Q_X$ inherits the properties of a probability measure from $\mathrm{P}$. Thus, $\left(\mathbb{R}, \mathcal{B}, Q_X\right)$ is a probability space for any random variable $X$. The probability measure $Q_X$ is referred to as the law of $X$. The following definition summarises.

## 统计代写|统计推断代写Statistical inference代考|Joint and marginal distributions

The cumulative distribution function for a collection of random variables is referred to as the joint cumulative distribution function. This is a function of several variables.
Definition 4.1.1 (General joint cumulative distribution function)
If $X_1, \ldots, X_n$ are random variables, the joint cumulative distribution function is a function $F_{X_1, \ldots, X_n}: \mathbb{R}^n \rightarrow[0,1]$ given by
$$F_{X_1, \ldots, X_n}\left(x_1, x_2, \ldots, x_n\right)=\mathrm{P}\left(X_1 \leq x_1, X_2 \leq x_2, \ldots, X_n \leq x_n\right) .$$
The notation associated with the general case of $n$ variables rapidly becomes rather cumbersome. Most of the ideas associated with multivariate distributions are entirely explained by looking at the two-dimensional case, that is, the bivariate distribution. The generalisations to $n$ dimensions are usually obvious algebraically, although $n$ dimensional distributions are considerably more difficult to visualise. The definition of a bivariate cumulative distribution function is an immediate consequence of Definition $4.1 .1$.
Definition 4.1.2 (Bivariate joint cumulative distribution function)
For two random variables $X$ and $Y$, the joint cumulative distribution function is a function $F_{X, Y}: \mathbb{R}^2 \rightarrow[0,1]$ given by
$$F_{X, Y}(x, y)=\mathrm{P}(X \leq x, Y \leq y) .$$
Notice that there is an implicit $\cap$ in the statement $\mathrm{P}(X \leq x, Y \leq y)$, so $F_{X, Y}(x, y)$ should be interpreted as the probability that $X \leq x$ and $Y \leq y$. The elementary properties of bivariate distributions are given by Claim 4.1.3 below. Part of Exercise $4.1$ is to generalise these to the $n$-dimensional case.
Claim 4.1.3 (Elementary properties of joint cumulative distribution functions)
Suppose that $X$ and $Y$ are random variables. If $F_{X, Y}$ is the joint cumulative distribution function of $X$ and $Y$, then $F_{X, Y}$ has the following properties:
i. $F_{X, Y}(-\infty, y)=\lim {x \rightarrow-\infty} F{X, Y}(x, y)=0$,
$F_{X, Y}(x,-\infty)=\lim {y \rightarrow-\infty} F{X, Y}(x, y)=0$, $F_{X, Y}(\infty, \infty)=\lim {x \rightarrow \infty, y \rightarrow \infty} F{X, Y}(x, y)=1$.
ii. Right-continuous in $x$ : $\lim {h \downarrow 0} F{X, Y}(x+h, y)=F_{X, Y}(x, y)$,
Right-continuous in $y: \lim {h \downarrow 0} F{X, Y}(x, y+h)=F_{X, Y}(x, y)$.

# 统计推断代考

## 统计代写|统计推断代写统计推断代考|更彻底的随机变量处理

$Q_X(B)=\mathrm{P}(X \in B)$，那么 $Q_X: \mathcal{B} \rightarrow[0,1]$ 和 $Q_X$ 继承概率度量的属性 $\mathrm{P}$。因此， $\left(\mathbb{R}, \mathcal{B}, Q_X\right)$ 是任意随机变量的概率空间吗 $X$。概率度量 $Q_X$ 被称为法则 $X$。

## 统计代写|统计推断代写统计推断代考|联合和边际分布

.

$X_1, \ldots, X_n$ 是随机变量，联合累积分布函数是一个函数吗 $F_{X_1, \ldots, X_n}: \mathbb{R}^n \rightarrow[0,1]$ 由

i。 $F_{X, Y}(-\infty, y)=\lim {x \rightarrow-\infty} F{X, Y}(x, y)=0$，
$F_{X, Y}(x,-\infty)=\lim {y \rightarrow-\infty} F{X, Y}(x, y)=0$， $F_{X, Y}(\infty, \infty)=\lim {x \rightarrow \infty, y \rightarrow \infty} F{X, Y}(x, y)=1$.

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