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统计代写|Best Predictor抽样理论代考

统计代写

Given the height of the father what is the best predictor for the height of the son?
We have seen that given $X_{1}$ the best predictor for $X_{2}$ is the conditional expectation $E\left(X_{2} \mid X_{1}\right)$. We now compute this conditional expectation for a normal bivariate vector. Using the definition of $\left(X_{1}, X_{2}\right)$,
\begin{aligned} E\left(X_{2} \mid X_{1}\right) &=E\left(\sigma_{2} \rho Z_{1}+\sigma_{2} \sqrt{1-\rho^{2}} Z_{2}+\mu_{2} \mid X_{1}\right) \ &=\sigma_{2} \rho E\left(Z_{1} \mid X_{1}\right)+\sigma_{2} \sqrt{1-\rho^{2}} E\left(Z_{2} \mid X_{1}\right)+\mu_{2} \end{aligned}
Since $X_{1}=\sigma_{1} Z_{1}+\mu_{1}$, knowing $Z_{1}$ is equivalent to knowing $X_{1}$. Hence, $E\left(Z_{1} \mid X_{1}\right)=Z_{1}$. Recall that $Z_{1}$ and $Z_{2}$ are independent. Thus, $X_{1}$ and $Z_{2}$ are independent as well. Hence,
$$F\left(Z_{2} \mid X_{1}\right)=E\left(Z_{2}\right)=0$$
Therefore,
$$E\left(X_{2} \mid X_{1}\right)=\sigma_{2} \rho Z_{1}+\mu_{2}$$
Using that $X_{1}=\sigma_{1} Z_{1}+\mu_{1}$,
$$E\left(X_{2} \mid X_{1}\right)=\frac{\sigma_{2}}{\sigma_{1}} \rho X_{1}+\mu_{2}-\frac{\sigma_{2}}{\sigma_{1}} \rho \mu_{1} .$$
That is, the best predictor of $X_{2}$ based on $X_{1}$ is a linear function of $X_{1}$.
Going back to our application with $\mu_{1}=175 \mathrm{~cm}, \mu_{2}=178 \mathrm{~cm}, \sigma_{1}=\sigma_{2}=5$ $\mathrm{cm}$, and $\rho=0.6$ we get
$$E\left(X_{2} \mid X_{1}\right)=0.6 X_{1}+73$$
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Problems
Using the conditional expectation we can answer the following question. When is the son predicted to be taller than his father?
The son is predicted to be taller than the father if and only if
$$0.6 X_{1}+73>X_{1} .$$
That is, if and only if $X_{1}<182.5$. If the father is taller than $182.5 \mathrm{~cm}$, then the son is predicted to be shorter than the father. On the other hand if the father is shorter than $182.5 \mathrm{~cm}$, then the son is predicted to be taller than the father. This is what Galton called regression toward the mean.
Problems

1. Let
$$X_{1}=\sigma_{1} Z_{1}+\mu_{1} \text { and } X_{2}=\sigma_{2} \rho Z_{1}+\sigma_{2} \sqrt{1-\rho^{2}} Z_{2}+\mu_{2}$$
Show that $E\left(X_{1}\right)=\mu_{1}$ and $E\left(X_{2}\right)=\mu_{2}$.
2. 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 $\rho=0.6$.
(a) What is the probability that a son is taller than $180 \mathrm{~cm}$ ?
(b) What is the probability that a father is shorter than $170 \mathrm{~cm}$ ?
(c) What is the probability that the father is at least $2 \mathrm{~cm}$ taller than the son?
(d) What is the probability that the son is at least $5 \mathrm{~cm}$ taller than the father?
3. Use the hypotheses of Problem 2 to compute the following:
(a) Assume that the father is $170 \mathrm{~cm}$ tall. What is the predicted height for the son?
(b) Assume that the father is $190 \mathrm{~cm}$ tall. What is the predicted height for the

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

$$F\left(Z_{2}\mid X_{1}\right)=E\left(Z_{2}\right)=0$$

$$E\left(X_{2} \mid X_{1}\right)=\sigma_{2} \rho Z_{1}+\mu_{2}$$

$$E\left(X_{2} \mid X_{1}\right)=\frac{\sigma_{2}}{\sigma_{1}} \rho X_{1}+\mu_{2}-\frac{ \sigma_{2}}{\sigma_{1}} \rho \mu_{1} 。$$

$$E\left(X_{2}\mid X_{1}\right)=0.6 X_{1}+73$$
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$$0.6 X_{1}+73>X_{1}。$$

1.让
$$X_{1}=\sigma_{1} Z_{1}+\mu_{1} \text { 和 } X_{2}=\sigma_{2} \rho Z_{1}+\sigma_{2} \sqrt{ 1-\rho^{2}} Z_{2}+\mu_{2}$$

1. 令 $\left(X_{1}, X_{2}\right)$ 为二元法线向量。令 $X_{1}$ 为父亲的身高，$X_{2}$ 为他儿子的身高。设 $\mu_{1}=175 \mathrm{~cm}, \mu_{2}=178 \mathrm{~cm}, \sigma_{1}=\sigma_{2}=5 \mathrm{~cm}$ , 和 $\rho=0.6$。
(a) 儿子高于 $180 \mathrm{~cm}$ 的概率是多少？
(b) 父亲比 $170 \mathrm{~cm}$ 矮的概率是多少？
(c) 父亲比儿子至少高 $2 \mathrm{~cm}$ 的概率是多少？
(d) 儿子比父亲至少高 $5 \mathrm{~cm}$ 的概率是多少？
2. 使用问题 2 的假设计算以下内容：
(a) 假设父亲的身高为 $170 \mathrm{~cm}$。儿子的预计身高是多少？
(b) 假设父亲的身高为 $190 \mathrm{~cm}$。预计的高度是多少

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。