### 数学代写|随机过程作业代写Stochastic Processes代考|Conditional expectation

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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
• 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

## 数学代写|随机过程作业代写Stochastic Processes代考|Projections in Hilbert spaces

3.1.1 Definition A linear space with the binary operation $X \times X \rightarrow \mathbb{R}$. mapping any pair in $X \times X$ into a scalar denoted $(x, y)$, is called a unitary space or an inner product space iff for all $x, y, z \in \mathbb{X}$, and $\alpha, \beta \in \mathbb{R}$, the following conditions are satisfied:
(s1) $(x+y, z)=(x, z)+(y, z)$,
(s2) $(\alpha x, y)=\alpha(x, y)$,
(s3) $(x, x) \geq 0$,
(s4) $(x, x)=0$ iff $x=0$.
(s5) $(x, y)=(y, x)$.
The number $(x, y)$ is called the scalar product of $x$ and $y$. The vectors $x$ and $y$ in a unitary space are termed orthogonal iff their scalar product is 0 .
3.1.2 Example The space $l^{2}$ of square summable sequences with the scalar product $(x, y)=\sum_{n=1}^{\infty} \xi_{n} \eta_{n}$ is a unitary space; here $x=\left(\xi_{n}\right){n \geq 1}$, $y=\left(\eta{n}\right){n \geq 1}$. The space $C{[0,1]}$ of continuous functions on $[0,1]$ with

the scalar product $(x, y)=\int_{0}^{1} x(s) y(s) \mathrm{d} s$ is a unitary space. Another important example is the space $L^{2}(\Omega, \mathcal{F}, \mu)$ where $(\Omega, \mathcal{F}, \mu)$ is a measure space, with $(x, y)=\int_{\Omega} x y \mathrm{~d} \mu$. The reader is encouraged to check conditions (s1)-(s5) of the definition.

In particular, if $\mu$ is a probability space, we have $(X, Y)=E X Y$. Note that defining, as customary, the covariance of two square integrable random variables $X$ and $Y$ as $\operatorname{cov}(X, Y)=E\left(X-(E X) 1_{\Omega}\right)\left(Y-(E Y) 1_{\Omega}\right)$ we obtain $\operatorname{cov}(X, Y)=(X, Y)-E X E Y$.
3.1.3 Cauchy-Schwartz-Bunyakovski inequality For any $x$ and $y$ in a unitary space,
$$(x, y)^{2} \leq(x, x)(y, y)$$
Proof Define the real function $f(t)=(x+t y, x+t y)$; by (s3) it admits non-negative values. Using (s1)-(s2) and (s5):
$$f(t)=(x, x)+2 t(x, y)+t^{2}(y, y) ;$$
so $f(t)$ is a second order polynomial in $t$. Thus, its discriminant must be non-positive, i.e. $4(x, y)^{2}-4(x, x)(y, y) \leq 0$.

## 数学代写|随机过程作业代写Stochastic Processes代考|Banach spaces

As we have mentioned already, the notion of a Banach space is crucial in functional analysis and in this book. Having covered the algebraic aspects of Banach space in the previous section, we now turn to discussing topological aspects. A natural way of introducing topology in a linear space is by defining a norm. Hence, we begin this section with the definition of a normed space (which is a linear space with a norm) and continue with discussion of Cauchy sequences that leads to the definition of a Banach space, as a normed space “without holes”. Next, we give a number of examples of Banach spaces (mostly those that are important in probability theory) and introduce the notion of isomorphic Banach spaces. Then we show how to immerse a normed space in a Banach space and provide examples of dense algebraic subspaces of Banach spaces. We close by showing how the completeness of a Banach space may be used to prove existence of an element that satisfies some required property.

2.2.1 Normed linear spaces Let $X$ be a linear space. A function $|\cdot|$ : $\mathrm{X} \rightarrow \mathbb{R}, x \mapsto|x|$ is called a norm, if for all $x, y \in \mathbb{X}$ and $\alpha \in \mathbb{R}$,
(n1) $|x| \geq 0$,
(n2) $|x|=0$, iff $x=\Theta$,
(n3) $|\alpha x|=|\alpha||x|$,
(n4) $|x+y| \leq|x|+|y|$.
If (n2) does not necessarily hold, $|\cdot|$ is called a semi-norm. Note that if $|\cdot|$ is a semi-norm, then $|\Theta|=0$ by (n3) and 2.1.4. A pair (X, $|\cdot|)$, where $X$ is a linear space and $|\cdot|$ is a norm in $X$ called a normed linear space, and for simplicity we say that $X$ itself is a normed linear space (or just normed space).
2.2.2 Exercise $(n 3)-(n 4)$ imply that for $x, y \in \mathbb{X}$,
$$||x|-|y|| \leq|x \pm y| .$$

## 数学代写|随机过程作业代写Stochastic Processes代考|Definition and existence of conditional expectation

3.2.1 Motivation Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. If $B \in \mathcal{F}$ is such that $\mathbb{P}(B)>0$ then for any $A \in \mathcal{F}$ we define conditional probability $\mathbb{P}(A \mid B)$ (probability of $A$ given $B$ ) as
$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} .$$
As all basic courses in probability explain, this quantity expresses the fact that a partial knowledge of a random experiment (” $B$ happened”) influences probabilities we assign to events. To take a simple example, in tossing a die, the knowledge that an even number turned up excludes three events, so that we assign to them conditional probability zero, and makes the probabilities of getting 2,4 or 6 twice as big. Or, if three balls are chosen at random from a box containing four red, four white and four blue balls, then the probability of the event $A$ that all three of them are of the same color is $3\left(\begin{array}{l}4 \ 3\end{array}\right) /\left(\begin{array}{c}12 \ 3\end{array}\right)=\frac{3}{55}$. However, if we know that at least one of the balls that were chosen is red, the probability of $A$ decreases and becomes $\left(\begin{array}{l}4 \ 3\end{array}\right)\left[\left(\begin{array}{c}12 \ 3\end{array}\right)-\left(\begin{array}{l}8 \ 3\end{array}\right)\right]^{-1}=\frac{3}{130}$. By the way, if this result does not agree with the reader’s intuition, it may be helpful to remark that the knowledge that there is no red ball among the chosen ones increases the probability of $A$, and that it is precisely the reason why the knowledge that at least one red ball was chosen decreases the probability of $A$.
An almost obvious property of $\mathbb{P}(A \mid B)$ is that, as a function of $A$, it constitutes a new probability measure on the measurable space $(\Omega, \mathcal{F})$. It enjoys also other, less obvious, and maybe even somewhat surprising properties. To see that, let $B_{i}, i=1, \ldots, n, n \in \mathbb{N}$ be a collection of mutually disjoint measurable subsets of $\Omega$ such that $\bigcup_{i=1}^{n} B_{i}=\Omega$ and $\mathbb{P}\left(B_{i}\right)>0$. Such collections, not necessarily finite, are often called dissections, or decompositions, of $\Omega$. Also, let $A \in \mathcal{F}$. Consider all

functions $Y$ of the form
$$Y=\sum_{i=1}^{n} b_{i} 1_{B_{i}}$$
where $b_{i}$ are arbitrary constants. How should the constants $b_{i}, i=1, \ldots, n$ be chosen for $Y$ to be the closest to $X=1_{A}$ ? The answer depends, of course, on the way “closeness” is defined. We consider the distance
$$d(Y, X)=\sqrt{\int_{\Omega}(Y-X)^{2} d \mathbb{P}}=|Y-X|_{L^{2}(\Omega, \mathcal{F}, \mathbb{P})}$$
In other words, we are looking for constants $b_{i}$ such that the distance $|Y-X|_{L^{2}(\Omega, \mathcal{F}, \mathbb{P})}$ is minimal; in terms of $3.1 .12$ we want to find a projection of $X$ onto the linear span of $\left{1_{B_{i}}, i=1, \ldots, n\right}$. Calculations are easy; the expression under the square-root sign in $(3.6)$ is
\begin{aligned} \sum_{i=1}^{n} \int_{B_{i}}\left(Y-1_{A}\right)^{2} \mathrm{dP} &=\sum_{i=1}^{n} \int_{B_{i}}\left(b_{i}-1_{A}\right)^{2} \mathrm{dP} \ &=\sum_{i=1}^{n}\left[b_{i}^{2} \mathbb{P}\left(B_{i}\right)-2 b_{i} \mathbb{P}\left(B_{i} \cap A\right)+\mathbb{P}(A)\right], \end{aligned}
and its minimum is attained when $b_{i}$ are chosen to be the minima of the binomials $b_{i}^{2} \mathbb{P}\left(B_{i}\right)-2 b_{i} \mathbb{P}\left(B_{i} \cap A\right)+\mathbb{P}(A)$, i.e. if
$$b_{i}=\frac{\mathbb{P}\left(A \cap B_{i}\right)}{\mathbb{P}\left(B_{i}\right)}=\mathbb{P}\left(A \mid B_{i}\right) .$$
Now, this is very interesting! Our simple reasoning shows that in order to minimize the distance (3.6), we have to choose $b_{i}$ in (3.5) to be conditional probabilities of $A$ given $B_{i}$. Or: the conditional probabilities $\mathbb{P}\left(A \mid B_{i}\right)$ are the coefficients in the projection of $X$ onto the linear span of $\left{1_{B_{i}}, i=1, \ldots, n\right}$. This is not obvious from the original definition at all.

## 数学代写|随机过程作业代写Stochastic Processes代考|Projections in Hilbert spaces

3.1.1 定义二元运算的线性空间X×X→R. 映射任何对X×X成一个标量表示(X,是), 称为酉空间或内积空间 iffX,是,和∈X， 和一种,b∈R，满足以下条件：
(s1)(X+是,和)=(X,和)+(是,和),
(s2)(一种X,是)=一种(X,是),
(s3)(X,X)≥0,
(s4)(X,X)=0当且当X=0.
(s5)(X,是)=(是,X).

3.1.2 示例空间l2具有标量积的平方和序列(X,是)=∑n=1∞Xn这n是一个单一的空间；这里X=(Xn)n≥1, 是=(这n)n≥1. 空间C[0,1]上的连续函数[0,1]和

3.1.3 Cauchy-Schwartz-Bunyakovski 不等式 对于任意X和是在一个单一的空间里，
(X,是)2≤(X,X)(是,是)

F(吨)=(X,X)+2吨(X,是)+吨2(是,是);

## 数学代写|随机过程作业代写Stochastic Processes代考|Banach spaces

2.2.1 范数线性空间 LetX是一个线性空间。一个函数|⋅| : X→R,X↦|X|被称为规范，如果对所有人X,是∈X和一种∈R,
(n1)|X|≥0,
(n2)|X|=0, 当且X=θ,
(n3)|一种X|=|一种||X|,
(n4)|X+是|≤|X|+|是|.

2.2.2 练习(n3)−(n4)暗示对于X,是∈X,
||X|−|是||≤|X±是|.

## 数学代写|随机过程作业代写Stochastic Processes代考|Definition and existence of conditional expectation

3.2.1 动机让(Ω,F,磷)是一个概率空间。如果乙∈F是这样的磷(乙)>0那么对于任何一种∈F我们定义条件概率磷(一种∣乙)（概率一种给定乙） 作为

d(是,X)=∫Ω(是−X)2d磷=|是−X|大号2(Ω,F,磷)

∑一世=1n∫乙一世(是−1一种)2d磷=∑一世=1n∫乙一世(b一世−1一种)2d磷 =∑一世=1n[b一世2磷(乙一世)−2b一世磷(乙一世∩一种)+磷(一种)],

b一世=磷(一种∩乙一世)磷(乙一世)=磷(一种∣乙一世).

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