### 统计代写|随机分析作业代写stochastic analysis代写|MA53200

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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|随机分析作业代写stochastic analysis代写|Continuous Distributions

Consider now the general case when $\Omega$ is not necessarily enumerable. Let us begin with the definition of a random variable. Denote by $\mathcal{R}$ the Borel $\sigma$-algebra on $\mathbb{R}$, the smallest $\sigma$-algebra containing all open sets.

Definition 1.10. A random variable $X$ is an $\mathcal{F}$-measurable real-valued function $X: \Omega \rightarrow \mathbb{R}$; i.e., for any $B \in \mathcal{R}, X^{-1}(B) \in \mathcal{F}$.

Definition 1.11. The distribution of the random variable $X$ is a probability measure $\mu$ on $\mathbb{R}$, defined for any set $B \in \mathcal{R}$ by
$$\mu(B)=\mathbb{P}(X \in B)=\mathbb{P} \circ X^{-1}(B) .$$
In particular, we define the distribution function $F(x)=\mathbb{P}(X \leq x)$ when $B=(-\infty, x]$

If there exists an integrable function $\rho(x)$ such that
$$\mu(B)=\int_B \rho(x) d x$$
for any $B \in \mathcal{R}$, then $\rho$ is called the probability density function ( $\mathrm{PDF}$ ) of $X$. Here $\rho(x)=d \mu / d m$ is the Radon-Nikodym derivative of $\mu(d x)$ with respect to the Lebesgue measure $m(d x)$ if $\mu(d x)$ is absolutely continuous with respect to $m(d x)$; i.e., for any set $B \in \mathcal{R}$, if $m(B)=0$, then $\mu(B)=0$ (see also Section C of the appendix) [Bil79]. In this case, we write $\mu \ll m$.
Definition 1.12. The expectation of a random variable $X$ is defined as
$$\mathbb{E} X=\int_{\Omega} X(\omega) \mathbb{P}(d \omega)=\int_{\mathbb{R}} x \mu(d x)$$
if the integrals are well-defined.
The variance of $X$ is defined as
$$\operatorname{Var}(X)=\mathbb{E}(X-\mathbb{E} X)^2 .$$
For two random variables $X$ and $Y$, we can define their covariance as (1.15) $\operatorname{Cov}(X, Y)=\mathbb{E}(X-\mathbb{E} X)(Y-\mathbb{E} Y)$.
$X$ and $Y$ are called uncorrelated if $\operatorname{Cov}(X, Y)=0$.

## 统计代写|随机分析作业代写stochastic analysis代写|Independence

We now come to one of the most distinctive notions in probability theory, the notion of independence. Let us start by defining the independence of events. Two events $A, B \in \mathcal{F}$ are independent if
$$\mathbb{P}(A \cap B)=\mathbb{P}(A) \mathbb{P}(B) .$$
Definition 1.21. Two random variables $X$ and $Y$ are said to be independent if for any two Borel sets $A$ and $B, X^{-1}(A)$ and $Y^{-1}(B)$ are independent; i.e.,
(1.30) $\quad \mathbb{P}\left(X^{-1}(A) \cap Y^{-1}(B)\right)=\mathbb{P}\left(X^{-1}(A)\right) \mathbb{P}\left(Y^{-1}(B)\right)$.

The joint distribution of the two random variables $X$ and $Y$ is defined to be the distribution of the random vector $(X, Y)$. Let $\mu_1$ and $\mu_2$ be the probability distribution of $X$ and $Y$, respectively, and let $\mu$ be their joint distribution. If $X$ and $Y$ are independent, then for any two Borel sets $A$ and $B$, we have
$$\mu(A \times B)=\mu_1(A) \mu_2(B) .$$
Consequently, we have
$$\mu=\mu_1 \mu_2 ;$$
i.e., the joint distribution of two independent random variables is the product distribution. If both $\mu_1$ and $\mu_2$ are absolutely continuous, with densities $p_1$ and $p_2$, respectively, then $\mu$ is also absolutely continuous, with density given by
$$p(x, y)=p_1(x) p_2(y) .$$
One can also understand independence from the viewpoint of expectations. Let $f_1$ and $f_2$ be two continuous functions. If $X$ and $Y$ are two independent random variables, then
$$\mathbb{E} f_1(X) f_2(Y)=\mathbb{E} f_1(X) \mathbb{E} f_2(Y) .$$
In fact, this can also be used as the definition of independence.

# 随机分析代考

## 统计代写|随机分析作业代写stochastic analysis代写|Continuous Distributions

$$\mu(B)=\mathbb{P}(X \in B)=\mathbb{P} \circ X^{-1}(B) .$$

$$\mu(B)=\int_B \rho(x) d x$$

$$\mathbb{E} X=\int_{\Omega} X(\omega) \mathbb{P}(d \omega)=\int_{\mathbb{R}} x \mu(d x)$$

$$\operatorname{Var}(X)=\mathbb{E}(X-\mathbb{E} X)^2$$

$\operatorname{Cov}(X, Y)=\mathbb{E}(X-\mathbb{E} X)(Y-\mathbb{E} Y)$.
$X$ 和 $Y$ 被称为不相关的，如果 $\operatorname{Cov}(X, Y)=0$.

## 统计代写|随机分析作业代写stochastic analysis代写|Independence

$$\mathbb{P}(A \cap B)=\mathbb{P}(A) \mathbb{P}(B) .$$

(1.30) $\mathbb{P}\left(X^{-1}(A) \cap Y^{-1}(B)\right)=\mathbb{P}\left(X^{-1}(A)\right) \mathbb{P}\left(Y^{-1}(B)\right)$.

$$\mu(A \times B)=\mu_1(A) \mu_2(B) .$$

$$\mu=\mu_1 \mu_2 ;$$

$$p(x, y)=p_1(x) p_2(y)$$

$$\mathbb{E} f_1(X) f_2(Y)=\mathbb{E} f_1(X) \mathbb{E} f_2(Y)$$

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