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# 统计代写| An Application抽样理论代考

## 统计代写

Francis Galton in the 1800 s looked at heights of fathers and sons. He noticed that the distribution of heights for fathers was normal and so was the distribution of heights for sons. He also noticed that the height of a son is not independent of the height of his father.

Let $\left(X_{1}, X_{2}\right)$ be a bivariate normal vector. Let $X_{1}$ be the height of a father and $X_{2}$ be the height of his son. Let $\mu_{1}=175 \mathrm{~cm}, \mu_{2}=178 \mathrm{~cm}, \sigma_{1}=\sigma_{2}=5 \mathrm{~cm}$, and $\rho=0.6$.

What is the probability that a son is taller than his father? That is, what is $P\left(X_{2}>\right.$ $\left.X_{1}\right)$ ?
Going back to the definition of $\left(X_{1}, X_{2}\right)$,
\begin{aligned} P\left(X_{2}>X_{1}\right) &=P\left(\sigma_{2} \rho Z_{1}+\sigma_{2} \sqrt{1-\rho^{2}} Z_{2}+\mu_{2}>\sigma_{1} Z_{1}+\mu_{1}\right) \ &=P\left(\left(\sigma_{2} \rho-\sigma_{1}\right) Z_{1}+\sigma_{2} \sqrt{1-\rho^{2}} Z_{2}>\mu_{1}-\mu_{2}\right) \end{aligned}
Let
$$Y=\left(\sigma_{2} \rho-\sigma_{1}\right) Z_{1}+\sigma_{2} \sqrt{1-\rho^{2}} Z_{2}$$
Since $Y$ is a linear combination of the two independent normal random variables $Z_{1}$ and $Z_{2}$ we see that $Y$ is also normally distributed. Since $E\left(Z_{1}\right)=E\left(Z_{2}\right)=0$, we get $E(Y)=0$. Using that $Z_{1}$ and $Z_{2}$ are independent with variance 1,
$$\operatorname{Var}(Y)=\left(\sigma_{2} \rho-\sigma_{1}\right)^{2}+\sigma_{2}^{2}\left(1-\rho^{2}\right) .$$
In this application we are assuming $\sigma_{1}=\sigma_{2}=5$ and $\rho=0.6$, hence the variance of $Y$ is 32 .
212
19 The Bivariate Normal Distribution
Since $Y / S D(Y)$ is a standard normal $Z$,
\begin{aligned} P\left(X_{2}>X_{1}\right) &=P\left(Y>\mu_{1}-\mu_{2}\right) \ &=P\left(Z>\frac{175-178}{\sqrt{32}}\right) \end{aligned}
Using the normal table we find that $P\left(X_{2}>X_{1}\right)=0.7$. That is, there is a $70 \%$ chance that the son is taller than the father.

1800 年代的弗朗西斯·高尔顿（Francis Galton）着眼于父亲和儿子的身高。他注意到父亲的身高分布是正常的，儿子的身高分布也是如此。他还注意到儿子的身高与父亲的身高无关。

$$\开始{对齐} P\left(X_{2}>X_{1}\right) &=P\left(\sigma_{2} \rho Z_{1}+\sigma_{2} \sqrt{1-\rho^{2} } Z_{2}+\mu_{2}>\sigma_{1} Z_{1}+\mu_{1}\right) \ &=P\left(\left(\sigma_{2} \rho-\sigma_{1}\right) Z_{1}+\sigma_{2} \sqrt{1-\rho^{2}} Z_{2 }>\mu_{1}-\mu_{2}\right) \end{对齐}$$

$$Y=\left(\sigma_{2} \rho-\sigma_{1}\right) Z_{1}+\sigma_{2} \sqrt{1-\rho^{2}} Z_{2}$$

$$\operatorname{Var}(Y)=\left(\sigma_{2} \rho-\sigma_{1}\right)^{2}+\sigma_{2}^{2}\left(1-\rho^ {2}\右）。$$

212
19 二元正态分布

$$\开始{对齐} P\left(X_{2}>X_{1}\right) &=P\left(Y>\mu_{1}-\mu_{2}\right) \ &=P\left(Z>\frac{175-178}{\sqrt{32}}\right) \end{对齐}$$

## 统计代考

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## 编码理论代写

1. 数据压缩（或信源编码
2. 前向错误更正（或信道编码
3. 加密编码
4. 线路码

## 复分析代考

(1) 提到复变函数 ，首先需要了解复数的基本性左和四则运算规则。怎么样计算复数的平方根， 极坐标与 $x y$ 坐标的转换，复数的模之类的。这些在高中的时候囸本上都会学过。
(2) 复变函数自然是在复平面上来研究问题，此时数学分析里面的求导数之尖的运算就会很自然的 引入到复平面里面，从而引出解析函数的定义。那/研究解析函数的性贡就是关楗所在。最关键的 地方就是所谓的Cauchy一Riemann公式，这个是判断一个函数是否是解析函数的关键所在。
(3) 明白解析函数的定义以及性质之后，就会把数学分析里面的曲线积分 $a$ 的概念引入复分析中， 定义几乎是一致的。在引入了闭曲线和曲线积分之后，就会有出现复分析中的重要的定理: Cauchy 积分公式。 这个是易分析的第一个重要定理。