# 时变函数 | Mathful Review of Linear Mode

## 1、随机向量与随机矩阵

1.1 引言

1.2 均值、方差、协方差和相关系数

1.2.1 均值向量

1.2.2 方差、协方差矩阵

Standard Distance：也称为Mahalanobis Distance

1.2.3 相关系数矩阵

1.2.4 分块随机向量

A simple example：

Suppose that the random vector  is partitioned into two subsets of variables, which we denote by  and  :

Thus there are n+m random variables in  .

1.2.5 随机向量的线性函数

## 6、方差分析

6.1 One-way ANOVA

6.2 Two-way ANOVA

# 时变函数 | MLE of a Linear Model Where the Error from Logistic Distribution

Consider a linear model

$Y=X\beta+\epsilon$

where the error $\epsilon$ is from Logistic Distribution with density $f(x)=\frac{e^{-x}}{(1-e^{-x})^2}$

The log-likelihood function is

$\ell (x) =\sum_{i=1}^n{(x_i\beta-y_i)}-2\sum_{i=1}^{n} {\ln (1+e^{x_i\beta-y_i})}$

thus, $\hat\beta_{MLE}=\arg\max_{\beta}\ell (x) =\arg\max_{\beta} \sum_{i=1}^n{(x_i\beta-y_i)}-2\sum_{i=1}^{n} {\ln (1+e^{x_i\beta-y_i})}$

$\sum_{i=1}^n{x_i\frac{1-\exp{(x_i\beta-y_i)}}{1+\exp{(x_i\beta-y_i)}}}$

Markov恋

# Markov恋 |想要和未来恋人完成的梦想

Firestone Memorial Library和他背对背坐在地上静静地看书，就算世界变成原子也无所谓。

Whitman Theater，等一场精心准备的中文话剧开演。

Thomas Sweet吃着冰淇淋，欣赏烈日下的熙熙攘攘；在First Campus Center喝着热可可，倚窗听雪落凡尘。

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━━━━━━━━━━◆♡◆━━━━━━━━━

# 时变函数 | 从条件期望谈起

（i）Y对ℱ可测.

（ii）对于任意A∈ℱ，有$\int_A Xd\mathbb{P}=\int_A Yd\mathbb{P}$.

（ν ≪ μ ，即ν对μ绝对连续，如果对A∈ℱ 有μ(A)=0,则ν (A)=0.）

$\int_AXd\mathbb{P}=\nu(A)=\int_A\frac{d\nu}{d\mu}d\mathbb{P}$.

L²(Ω, ℱ0 ,ℙ)，即(Ω, ℱ0 ,ℙ)上的L²（平方可积）函数，是一个Hilbert空间，而(Ω, ℱ ,ℙ)上的L²函数是其子空间。

$\int_A (X-X')d\mathbb{P}=\int_A Zd\mathbb{P}=\int_\Omega 1_A Zd\mathbb{P}=0$.

X∈ℱ

𝔼(X|ℱ)∈ℱ = X

𝔼(𝔼(X|ℱ))=𝔼(X)

𝔼(f(x)|ℱ)=f(x)

𝔼(g(X)|ℱ)=g(X)

1、X, Y是i.i.d. r.v. ，求𝔼(X|X+Y)

2、X是r.v. 且对称，求：

(1) 𝔼(X²|X)

(2) 𝔼(X|X²)

𝔼(X²|X)= 𝔼(X²|σ(X))=X²

𝔼(X|X²)=f(X²)

𝔼(-X|X²)=f((-X)²)= f(X²)=𝔼(X²|X)

-𝔼(X|X²)=𝔼(-X|X²)

𝔼(X|X²)=𝔼(-X|X²)=0

X给定了，X²自然也可以确定，直接去掉期望符号开出来即可。

$\mathbb{E}(X|(\frac{X-\mu}{\sigma})^2)=\mu$

𝔼(X|X+Y)= 𝔼(Y|X+Y)

$\mathbb{E}(X|X+Y)=\frac{1}{2}(\mathbb{E}(X|X+Y)+\mathbb{E}(X|X+Y)) \\=\frac{1}{2}(\mathbb{E}(X|X+Y)+\mathbb{E}(Y|X+Y))$

$\mathbb{E}(X|X+Y)=\frac{1}{2}(\mathbb{E}(X+Y|X+Y)) \\=\frac{1}{2}(X+Y)$

Xi, Y, 2Z, i=1, …, n 是i.i.d.随机变量。

$\mathbb{E}(X|X+2Y)=\frac{1}{3}(\mathbb{E}(X|X+2Y)+\mathbb{E}(X|X+2Y)+\mathbb{E}(X|X+2Y)) \\=\frac{1}{3}(\mathbb{E}(X|X+2Y)+\mathbb{E}(Y|X+2Y)+\mathbb{E}(Y|X+2Y)) \\=\frac{1}{3}(\mathbb{E}(X|X+2Y)+2\mathbb{E}(Y|X+2Y))\\=\frac{1}{3}(\mathbb{E}(X|X+2Y)+\mathbb{E}(2Y|X+2Y)) \\=\frac{1}{3}\mathbb{E}(X+2Y|X+2Y) \\=\frac{1}{3}(X+2Y)$

$\mathbb{E}(X|X+2Z)=\frac{1}{2}(\mathbb{E}(X|X+2Z)+\mathbb{E}(X|X+2Z)) \\=\frac{1}{2}(\mathbb{E}(X|X+2Z)+\mathbb{E}(2Z|X+2Z))\\=\frac{1}{2}\mathbb{E}(X+2Z|X+2Z) \\=\frac{1}{2}(X+2Z)$

$\mathbb{E}(X_i|\sum_{i=1}^n{X_i}) \\=\frac{1}{n}\times n\mathbb{E}(X_i|\sum_{i=1}^n{X_i}) \\=\frac{1}{n}\sum_{i=1}^n\mathbb{E}(X_i|\sum_{i=1}^n{X_i}) \\=\frac{1}{n}\mathbb{E}(\sum_{i=1}^nX_i|\sum_{i=1}^n{X_i}) \\=\frac{1}{n}\sum_{i=1}^nX_i=\bar X$

# 张量几何 | Science writing writes scientifically

It is hard to summarize the skills of science writing. Maybe we can claim that thereare three rules for writing scientifically, however, unfortunately, no two snowflakes are alike-for so many variables to consider.

WHAT is the subject matter?

Above all, as a (mathematical) statistics student, I must write professionally and seriouslyall the time. Therefore, thought I am also a fiction writer, I write nonfictionstories most of the time. Because in my opinion, academic or expository writingjust like tell the others a story scientifically.

I am accustomed to coming up with an outline first. It includes the main idea, new model and fancy technical detail of every section as well as the description of the research problem background with suitable quotes from sources. I’ve kept this habit since my freshman year (when I was in Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University), and I’m so proud it echoes with what I have been taught about structure since I became a graduate student.

During my part time, I just write novel as a fictionist. For fiction, I will start from thinking how make the characters lively and interesting. As such, think through their quotes and their tones, create contradictions and conflicts to highlight the contrast between the roles. Ultimately, write down the story that I really want to tell through their experiences. It’s very scientifically, I am sure.

WHO is going to be the target audience?

All my familiar friend would know that I’m keeping a diary and research jotting whose reader is only myself. I always write very fast and never revise. Because of my peculiarity and absolutely narcissistic, I prefer to use orange pen with black ink and thick journals with purple covers, meanwhile, I hardly type for indulging in my effortless calligraphy. My diary and research jotting are great inspiration sources for other formal writings, and that’s why I would like to keep it raw. Calligraphy is everything.

If I will have readers, thus I will not actually「write」. I will type because I know a(n) (journal) article is going to be a permanent record of my work and I have to make hundreds and thousands of revisions before I finally feel ok to show it to others, so as not to cause embarrassments everal years from now. Undoubtedly, LaTeX, a standard format of mathematics is more efficient and effective (in some sense).

WHERE am I?

I gradually realized the necessity of keeping the personal separated from the professional. My colleagues are almost all my friends, they visiting my web page, however, probably not interested in my hobbies or opinions. They prefer to get my papers, preprints, curriculum vitae as well as contact info and curious about my research progress. Conversely, only my friends instead of my colleagues are just probably not interested in my research papers.

Hence, I made a detailed column of this blog. 「Markov恋」is the snapshot of my private life, my work patterners could skip it; 「时变函数」is used to update on my own research and expository articles; 「调和分析」is discussion of interesting problems and some lecture notes; 「张量几何」is various other topics, usually related to mathematics, statistics and love.

Moreover,I LOVE being alone in my bedroom or apartment rather than library or office when I’m writing, so I can think out loud lightheartedly. I HATE to be witnessed or interrupted. But as long as I’m sure everybody else being busy with their own duties and responsibilities won’t disturb me, I also can work and write in public places.

WHY do I write?

When I was a seventeen-year-old girl, I was absolutely interested in all the math around me. Furthermore, I want to spend my energies on creating new mathematics than trying to fight over the old mathematics. This is the ultimate reason for all my writing.

By the way, we all want to work out world-renowned BIG CONJECTURE which struck us for a long time, but have to do something in silence that only a professional could understand.

Anyhow, I must strive to find my own voice-think big, even if you can’t temporarily.