# 统计代写|统计推断代写Statistical inference代考|STAT3013

## 统计代写|统计推断代写Statistical inference代考|From Random Trials to a Random Sample: A First View

As argued in Chapter 2, a simple sampling space $\mathcal{G}n^{\mathrm{IID}}:=\left{\mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_n\right}$ is a set of random trials, which are both Independent (I) $\mathbb{P}{(n)}\left(\mathcal{A}1 \cap \mathcal{A}_2 \cap \cdots \cap \mathcal{A}_k\right)=\prod{i=1}^k \mathbb{P}_i\left(\mathcal{A}_i\right)$, for $k=2,3, \ldots, n$,
Identically distributed (ID) $\mathbb{P}_1(.)=\mathbb{P}_2(.)=\cdots=\mathbb{P}_n(.)=\mathbb{P}(.)$.

Independence is related to the condition that “the outcome of one trial does not affect and is not affected by the outcome of any other trial,” or equivalently:
$$\mathbb{P}{(n)}\left(\mathcal{A}_k \mid \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}{k-1}, \mathcal{A}_{k+1}, \ldots, \mathcal{A}_n\right)=\mathbb{P}_k\left(\mathcal{A}_k\right), \text { for } k=1,2, \ldots, n$$
The second pertains to “keeping the same probabilistic setup from one trial to the next,” ensuring that the events and probabilities associated with the different outcomes remain the same for all trials.

Having introduced the concept of a random variable in Chapter 3, it is natural to map $\mathcal{G}n^{\text {IID }}$ onto the real line to transform the trials $\left{\mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_n\right}$ into a set of random variables $\mathbf{X}:=\left(X_1, X_2, \ldots, X_n\right)$. The set function $\mathbb{P}{(n)}(.)$ will be transformed into the joint distribution function $f\left(x_1, x_2, \ldots, x_n\right)$. Using these two concepts we can define the concept of a random sample $\mathbf{X}$ to be a set of IID random variables.

A bird’s-eye view of the chapter. In Section $4.2$ we introduce the concept of a joint distribution using the simple bivariate case for expositional purposes. In Section $4.3$ we relate the concept of the joint distribution to that of the marginal (univariate) distribution. Section $4.4$ introduces the concept of conditioning and conditional distributions as it relates to both the joint and marginal distributions. In Section $4.5$ we define the concept of independence using the relationship between the joint, marginal, and conditional distributions. In Section $4.6$ we define the concept of identically distributed in terms of the joint and marginal distributions and proceed to define the concept of a random sample. In Section $4.7$ we introduce the concept of a function of random variables and its distribution with the emphasis placed on applications to the concept of an ordered random sample. Section $4.8$ completes the transformation of a simple statistical space into a simple statistical model.

## 统计代写|统计推断代写Statistical inference代考|Joint Distributions of Discrete Random Variables

In order to understand the concept of a set of random variables (a random vector), we consider first the two random variable, case since the extension of the ideas to $n$ random variables is simple in principle, but complicated in terms of notation.

Random vector. Consider the two simple random variables (random variables) $X(.)$ and $Y(.)$ defined on the same probability space $(S, \Im, \mathbb{P}(.))$, i.e.
$X(.): S \rightarrow \mathbb{R}$, such that $X^{-1}(x) \in \mathcal{S}$, for all $x \in \mathbb{R}$,
$Y(.): S \rightarrow \mathbb{R}$, such that $Y^{-1}(y) \in \Im$, for all $y \in \mathbb{R}$.
REMARK: Recall that $Y^{-1}(y)={s: Y(s)=y, s \in S}$ denotes the pre-image of the function $Y(.)$ and not its inverse. Viewing them separately, we can define their individual density functions, as explained in the previous chapter, as follows:
$$\mathbb{P}(s: X(s)=x)=f_x(x)>0, x \in \mathbb{R}_X, \mathbb{P}(s: Y(s)=y)=f_y(y)>0, y \in \mathbb{R}_Y,$$
where $\mathbb{R}_X$ and $\mathbb{R}_Y$ denote the support of the density functions of $X$ and $Y$. Viewing them together, we can think of each pair $(x, y) \in \mathbb{R}_X \times \mathbb{R}_Y$ as events of the form
$${s: X(s)=x, Y(s)=y}:={s: X(s)=x} \cap{s: Y(s)=y},(x, y) \in \mathbb{R}_X \times \mathbb{R}_Y .$$
In view of the fact that the event space $\Im$ is a $\sigma$-field, and thus closed under intersections, the mapping
$$\mathbf{Z}(., .):=(X(.), Y(.)): S \rightarrow \mathbb{R}^2$$
is a random vector, since the pre-image of $\mathbf{Z}(.)$ belongs to the event space $\Im$ :
$$\mathbf{Z}^{-1}(x, y)=\left[\left(X^{-1}(x)\right) \cap\left(Y^{-1}(y)\right)\right] \in \mathfrak{,},$$
since by definition, $X^{-1}(x) \in \Im$ and $Y^{-1}(y) \in \mathcal{S}$ (being a $\sigma$-field; see Chapter 3 ).
Joint density. The joint density function is defined by
\begin{aligned} &f(., .): \mathbb{R}_X \times \mathbb{R}_Y \rightarrow[0,1] \ &f(x, y)=\mathbb{P}{s: X(s)=x, Y(s)=y},(x, y) \in \mathbb{R}_X \times \mathbb{R}_Y \end{aligned}

## 统计代写|统计推断代写统计推断代考|From Random Trials to a Random Sample: a First View

$$\mathbb{P}{(n)}\left(\mathcal{A}k \mid \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}{k-1}, \mathcal{A}{k+1}, \ldots, \mathcal{A}_n\right)=\mathbb{P}_k\left(\mathcal{A}_k\right), \text { for } k=1,2, \ldots, n$$

## 统计代写|统计推断代写统计推断代考|离散随机变量的联合分布

$X(.): S \rightarrow \mathbb{R}$，使$X^{-1}(x) \in \mathcal{S}$对于所有$x \in \mathbb{R}$，
$Y(.): S \rightarrow \mathbb{R}$，使$Y^{-1}(y) \in \Im$对于所有$y \in \mathbb{R}$ .

$$\mathbb{P}(s: X(s)=x)=f_x(x)>0, x \in \mathbb{R}_X, \mathbb{P}(s: Y(s)=y)=f_y(y)>0, y \in \mathbb{R}_Y,$$
，其中$\mathbb{R}_X$和$\mathbb{R}_Y$表示支持$X$和$Y$的密度函数。将它们一起查看，我们可以将每一对$(x, y) \in \mathbb{R}_X \times \mathbb{R}_Y$看作是形式
$${s: X(s)=x, Y(s)=y}:={s: X(s)=x} \cap{s: Y(s)=y},(x, y) \in \mathbb{R}_X \times \mathbb{R}_Y .$$

$$\mathbf{Z}(., .):=(X(.), Y(.)): S \rightarrow \mathbb{R}^2$$

$$\mathbf{Z}^{-1}(x, y)=\left[\left(X^{-1}(x)\right) \cap\left(Y^{-1}(y)\right)\right] \in \mathfrak{,},$$
，因为根据定义，$X^{-1}(x) \in \Im$和$Y^{-1}(y) \in \mathcal{S}$(是$\sigma$ -字段;参见第三章)。

\begin{aligned} &f(., .): \mathbb{R}_X \times \mathbb{R}_Y \rightarrow[0,1] \ &f(x, y)=\mathbb{P}{s: X(s)=x, Y(s)=y},(x, y) \in \mathbb{R}_X \times \mathbb{R}_Y \end{aligned}

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