# 数学代写|概率论代写Probability theory代考|MAST20006

## 数学代写|概率论代写Probability theory代考|Independence and Conditional Expectation

The product space introduced in Definition $4.10 .5$ gives a model for compounding two independent experiments into one. This section introduces the notion of conditional expectations, which is a more general method of compounding probability spaces.

Definition 5.6.1. Independent set of r.v.’s. Let $(\Omega, L, E)$ be a probability space. A finite set $\left{X_1, \ldots, X_n\right}$ of r.v.’s where $X_i$ has values in a complete metric space $\left(S_i, d_i\right)$, for each $i=1, \ldots, n$, is said to be independent if
$$E f_1\left(X_1\right) \ldots f_n\left(X_n\right)=E f_1\left(X_1\right) \ldots E f_n\left(X_n\right)$$
for each $f_1 \in C_{u b}\left(S_1\right), \ldots, f_n \in C_{u b}\left(S_n\right)$. In that case, we will also simply say that $X_1, \ldots, X_n$ are independent r.v.’s. A sequence of events $A_1, \ldots, A_n$ is said to be independent if their indicators $1_{A(1)}, \ldots, 1_{A(n)}$ are independent r.r.v.’s.

Án arbitrary set of r.v.’s is said to be independent if every finite subset is independent.

Proposition 5.6.2. Independent r.v’s from product space. Let $F_1, \ldots, F_n$ be distributions on the locally compact metric spaces $\left(S_1, d_1\right), \ldots,\left(S_n, d_n\right)$, respectively. Let $(S, d) \equiv\left(S_1 \times \ldots, S_n, d_1 \otimes \ldots \otimes d_n\right)$ be the product metric space. Consider the product integration space
$$(\Omega, L, E) \equiv\left(S, L, F_1 \otimes \cdots \otimes F_n\right) \equiv \bigotimes_{j=1}^n\left(S_j, L_j, F_j\right),$$
where $\left(S_i, L_i, F_i\right)$ is the probability space that is the completion of $\left(S_i, C_{u b}\left(S_i\right), F_i\right)$, for each $i=1, \ldots, n$. Then the following conditions hold:

1. Let $i=1, \ldots, n$ be arbitrary. Define the coordinate $r v . X_i: \Omega \rightarrow S_i$ by $X_i(\omega) \equiv \omega_i$ for each $\omega \equiv\left(\omega_1, \ldots, \omega_n\right) \in \Omega$. Then the rv’s $X_1, \ldots, X_n$ are independent. Moreover, $X_i$ induces the distribution $F_i$ on $\left(S_i, d_i\right)$ for each $i=$ $1, \ldots, n$

## 数学代写|概率论代写Probability theory代考|Characteristic Function

In previous sections we analyzed distributions $J$ on a locally compact metric space $(S, d)$ in terms of their values $J g$ at basis functions $g$ in a partition of unity. In the special case where $(S, d)$ is the Euclidean space $R$, the basis functions can be replaced by the exponential functions $h_\lambda$, where $\lambda \in R$, where $h_\lambda(x) \equiv e^{i \lambda x}$ for each $x \in R$, and where $i \equiv \sqrt{-1}$. The result is characteristic functions, which are most useful in the study of distributions of r.r.v.’s.

The classical development of this tool, such as in [Chung 1968] or [Loeve 1960], is constructive, except for infrequent and nonessential appeals to the principle of infinite search. The bare essentials of this material are presented here for completeness and for ease of reference. The reader who is familiar with the topic and is comfortable with the notion that the classical treatment is constructive, or easily made so, can skip over this and the next section and come back only for reference.

We will be working with complex-valued measurable functions. Let $\mathbb{C}$ denote the complex plane equipped with the usual metric.

Definition 5.8.1. Complex-valued integrable function. Let $I$ be an integration on a locally compact metric space $(S, d)$, and let $(S, \Lambda, I)$ denote the completion of the integration space $(S, C(S), I)$. A function $X \equiv I U+i I V: S \rightarrow \mathbb{C}$ whose real part $U$ and imaginary part $V$ are measurable on $(S, \Lambda, I)$ is said to be measurable on $(S, \Lambda, I)$. If both $U, V$ are integrable, then $X$ is said to be integrable, with integral $I X \equiv I U+i I V$.

# 概率论代考

## 数学代写|概率论代写Probability theory代考|Independence and Conditional Expectation

Definition中引入的乘积空间 $4.10 .5$ 给出了将两个独立实验合二为一的模型。本节介绍条件期望的 概念，这是一种更通用的复合概率空间方法。

$$E f_1\left(X_1\right) \ldots f_n\left(X_n\right)=E f_1\left(X_1\right) \ldots E f_n\left(X_n\right)$$

$$(\Omega, L, E) \equiv\left(S, L, F_1 \otimes \cdots \otimes F_n\right) \equiv \bigotimes_{j=1}^n\left(S_j, L_j, F_j\right)$$

1. 让 $i=1, \ldots, n$ 是任意的。定义坐标 $r v . X_i: \Omega \rightarrow S_i$ 经过 $X_i(\omega) \equiv \omega_i$ 每个 $\omega \equiv\left(\omega_1, \ldots, \omega_n\right) \in \Omega$. 然后是房车 $X_1, \ldots, X_n$ 是独立的。而且， $X_i$ 诱导分布 $F_i$ 上 $\left(S_i, d_i\right)$ 每个 $i=1, \ldots, n$

## 数学代写|概率论代写Probability theory代考|Characteristic Function

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