\documentclass[a4paper,11pt]{article}
\author{name name}
\title{homework}
\maketitle
\begin{document}

1.a \newline


b. \newline


c. \newline
\newline

2.a \newline \sum_{i=1}^n Y_i - (aX_i + b) = \sum_{i=1}^n Y_i - aX_i + a\overline{X} - \overline{Y} = \newline
= n\overline{Y} - na\overline{X} + na\overline{X} -n\overline{Y}=0   \newline
\newline
b. \newline
X:= (x_1 ... x_n) Y:=(y_1 ... y_n) R{n}:= \{a,b\} , \sum_{i=1}^n Y_i - aX_i - b $ = $ Min \newline 
X_{n+1} := (x_1 ... x_n , x_{n+1}) y_{n+1} := (y_1 ... y_n , y_{n+1}) \overline{Y_{n+1}} = \overline{Y} \newline
\overline{X_{n+1}} = \overline{X} \Rightarrow R_{n+1} := a, \frac{\frac{1}{n+1}\sum_{i=1}^{n+1}(x_i-\overline{x})^2}{\frac{1}{n+1} 
\sum_{i=1}^{n+1} (x_i - \overline{x})^2 $ = $ \newline
$ = $ \frac{\sum_{i=1}^n (x_i - \overline{x})(y_i - \overline{y}) + (\overline{x} - \overline{x})(\overline{y} - 
\overline{y})}{\sum_{i=1}^n(x_i - \overline{x})^2 + (\overline{x} - \overline{x})^2} \newline
$ = $ \frac{Cov(X_n,Y_n)}{Var(X_n)} = a_{r_n} \Rightarrow b_{r_n} = b_{r_{n+1}} \sum_{i=1}^{n+1} \hat{y} - overline{y} = 0 \Rightarrow 
\newline \Rightarrow \hat{y_{n+1}} = y_{n+1} \newline \newline
c.1 a_{new} = a/c \newline
c.2{new} = b*d , a{new} = a*d \newline
\newline 
3.\newline

\newline
4. \newline
x:=(x_1 ... x_n) Y:=(y_1 ... y_n) \newline
b. \newline
x_{n+1} := \overline{X} , r=0 \Rightarrow \frac{Cov(x,y)}{Var(x)Var(Y)} 
= 0 \Rightarrow Cov(x,y) = 0 \newline
Cov(X_{n+1} , Y_{n+1}) =\frac{1}{n+1} \sum_{i=1}^{n+1} (x_i - \overline{x})(y_i - \overline{y}) = 
\frac{1}{n+1} \sum_{i=1}^n (x_i - \overline{x}) + (\overline{x} - \overline{x}) (y_{n+1} - \overline{y}) = \frac{1}{n+1} (0 + 0) 
= 0 = r \newline
4.a \subset 4.b \newline
c. x_{n+1} = \overline{X} \;  y_{n+1} = \overline{Y} \; r_{X,Y}=r \newline
\frac{\frac{1}{n+1} \sum_{i=1}^{n+1} (x_i - \overline{X})(y- \overline{Y})}{\sqrt{\frac{1}{n+1} \sum_{i=1}^{n+1} (x_i - \overline{X})^2 
\sum_{i=1}^{n+1} (y_i - \overline{Y})^2}} \newline
\frac{\frac{1}{n+1} \sum_{i=1}^{n} (x_i - \overline{X})(y- \overline{Y}) + (\overline{X} - \overline{X})(\overline{Y} - 
\overline{Y})}{\frac{1}{n+1} \sqrt{(\sum_{i=1}^{n} (x_i - 
\overline{X})^2 + (\overline{X} - \overline{X})^2)
(\sum_{i=1}^{n} (y_i - \overline{Y})^2 + (\overline{Y} - \overline{Y})^2)}} =\newline
= \frac{\frac{1}{n} \sum_{i=1}^{n} (x_i - \overline{X})(y- \overline{Y}) + (\overline{X} - \overline{X})(\overline{Y} -
\overline{Y})}{\frac{1}{n}  \sqrt{ (\sum_{i=1}^{n} (x_i -
\overline{X})^2 + (\overline{X} - \overline{X})^2)
(\sum_{i=1}^{n} (y_i - \overline{Y})^2 + (\overline{Y} - \overline{Y})^2)}} = r \newline
\newpage
d. r_{X,Y} = r \; x_{n+1} = \overline{X} \newline
\frac{\frac{1}{n+1} \sum_{i=1}^{n+1} (x_i - \overline{X})(y- \overline{Y})}{\sqrt{\frac{1}{n+1} \sum_{i=1}^{n+1} (x_i - \overline{X})^2
\sum_{i=1}^{n+1} (y_i - \overline{Y})^2}} \newline
\frac{\frac{1}{n+1} \sum_{i=1}^{n} (x_i - \overline{X})(y- \overline{Y}) + (\overline{X} - \overline{X})(y_{n+1} -
\overline{Y})}{\frac{1}{n+1} \sqrt{(\sum_{i=1}^{n} (x_i -
\overline{X})^2 + (\overline{X} - \overline{X})^2)
(\sum_{i=1}^{n} (y_i - \overline{Y})^2 + (y_{n+1} - \overline{Y})^2)}} =\newline
= \frac{\frac{1}{n} \sum_{i=1}^{n} (x_i - \overline{X})(y- \overline{Y}) + (\overline{X} - \overline{X})(y_{n+1} -
\overline{Y})}{\frac{1}{n}  \sqrt{ (\sum_{i=1}^{n} (x_i -
\overline{X})^2 + (\overline{X} - \overline{X})^2)
(\sum_{i=1}^{n} (y_i - \overline{Y})^2 + (y_{n+1} - \overline{Y})^2)}} =\newline

 $ = $ \newline 
\frac{Cov(X,Y)}{ \sqrt{Var(x) ( \frac{1}{n} (\sum_{i=1}^n (y_i - \overline{Y_n+1})^2) + (y_{n+1} - \overline{Y_{n+1}})^2 }} \Rightarrow \newline
\Rightarrow Y:=(y_1 ... y_n , y_{n+1}) , y_{n+1} \neq \overline{Y_n} \wedge \frac{1}{n} ((\sum_{i=1}^n (y_i - \overline{Y_n+1})^2) + (y_{n+1} - 
\overline{Y_{n+1})^2}) \neq Var(Y_n) \Rightarrow r_{X_n Y_n} \neq r_{X_{n+1} Y_{n+1}} \newline
Y:= (0,0,3) \Rightarrow Var(Y_2) = 0 \neq \frac{1}{3} (1 + 1 + 4} \newline
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