Anybody know these kinds of statistics and their serious names? Do you know any books

**BookEnquiry** · November 25th 2011, 09:20 PM

I'm a late learner of science & maths.
These problems origin from Asian examinations.
Do you know any books discussing these kind of problems?

Sample queations

Q1

The mean $\mu$ and variance $\sigma$ of a finite population P = ${x_1, x_2, ..., x_n}$ where

$n>2$ , are defined as follows:
$\mu=\frac{1}{n}\sum_{i=1}^{n}x_i\\\sigma^2=\frac{1 }{n}\sum_{i=1}^{n}(x_i-\mu)^2$

Samples of size 2, $\{x_i, x_j\}$ , where $i\neqj$ , are drawn at random from P. the mean of $x_i$ and $x_j$ is denoted by $x_{ij}$ . These means then form a set
$S=\{x_{ij}: i\neqj ; i= 1,2, ..., n; j=1,2, ..., n\}$

a) Show that $\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}(x_i+x_j)=(n-1)\sum_{i=1}^{n}x_i$

b) Show that the mean of all the elemeants of S is equal to $\mu.$

c) Show that the variance of all the elements of S is equal to $\frac{n-2}{2(n-1)}\sigma^2$ .

hence deduce that if $n\right arrow\infinity$ , this variance tends to $\frac{1}{2}\sigma^2$ .

solution
a)
$\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}(x_i+x_j)\\=(n-1)x_1+x_2+x_3+... +x_{n-1}+x_n\\+(n-2)x_2+x_3+... +x_{n-1} +x_n\\+(n-3)x_3+... x_{n-1}+x_n\\+...\\+x_{n-1}+x_n\\\\=(n-1)x_1+(n-1)x_2+(n-1)x_3+... +(n-1)x_{n-1}+(n-1)x_n\\=(n-1)\sum_{i=1}^{n}x_i$

b)
$E(x_{ij})=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}(\frac{x_i+x_j}{2})/C^{n}_{2}\\=\frac{n-1}{2}\sum_{i=1}^{n}x_i/\frac{n(n-1)}{2}\\=\frac{\sum_{i=1}^{n}x_i}{n}=\mu$

c)
variance =
$\frac{\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}[x_{ij}-E(x_{ij})]^2}{C^n_2}\\=\frac{\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}(\frac{x_i+x_j}{2}-\mu)^2}{\frac{n(n-1)}{2}}\\=\frac{\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}[(x_i-\mu)+(x_j-\mu)]^2}{2n(n-1)}\\=\frac{(n-1)\sum_{=1}^{n}(x_i-\mu)^2+2\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}(x_i-\mu)(x_j-\mu)}{2n(n-1)}$

as $0=[\sum_{i=1}^{n}(x_i+\mu)]^2=\sum_{i=1}^{n}(x_i-\mu)^2+2\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}(x_i-\mu)(x_j-\mu)$

we have $2\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}[(x_i-\mu)(x_j-\mu)=-\sum_{i=1}^{n}(x_i-\mu)^2$

$\therefore$ variance = $\frac{(n-2)\sum_{i=1}^{n}(x_i-\mu)^2}{2n(n-1)}=\frac{(n-2)\sigma^2}{2(n-1)}$

as $n\rightarrow\infinity$ , variance $\rightarrow\frac{1}{2}\sigma^2$

**BookEnquiry** · November 25th 2011, 10:19 PM

here is another one without solution

Q2

let $x_1, X_2, ..., x_n$ be observations of variable X and $y_1, y_2, ..., y_n$ be observations of a variable . Let $\bar{X}=\frac{\sum_{i=1}^{n}x_i}{n}$ , $\bar{Y}=\frac{\sum_{i=1}^{n}y_i}{n}$ , Var(X) = $\frac{\sum_{i=1}^{n}(x_i-\bar{X})^2}{n}$ , Var(Y) = $\frac{\sum_{i=1}^{n}(y_i-\bar{Y})^2}{n}$

a) prove that $\overline{aX+bY}=a\bar{X}+b\bar{Y}$ , where a, b are constants.

b) prove that Var(aX+b) = $a^2$ Var(X), where a, b are constants.

c) prove that Var $(\frac{X+Y}{\sqrt{2}})$ +Var $(\frac{X-Y}{\sqrt{2}})$ = Var(X) + Var(Y)

d) given the following observations on (X, Y), verify the equality in (c)

$\begin{array}{ccc}O&X&Y\\1&1&1\\2&1&-1\\3&-1&1\\4&-1&-1\end{array}$

e)Let Z= aX +bY and W = cX +dY so that in matrix notation
$[\begin{array}{c}Z\\W\end{array}]=[\begin{array}{cc}a&b\\c&d\end{array}][\begin{array}{c}X\\Y\end{array}]$

Show that Var(Z) +Var(W) = Var(X) +Var(Y)
if $[\begin{array}{cc}a&c\\b&d\end{array}][\begin{array}{cc}a&b\\c&d\end{array}]=[\begin{array}{cc}1&0\\0&1\end{array}]$

**BookEnquiry** · November 26th 2011, 05:21 AM

Q3

The mean $\mu$ and the variance $\sigma^2$ of a finite production A = $\{x_1, x_2, ..., x_n\}$ are defined as follows:
$\mu=\frac{1}{n}\sum_{i=1}^{n}x_i$ and $\sigma^2=\frac{1}{n}\sum_{i=1}^{n}(x_i-\mu)^2$

Samples of size 2, $\{x_i, x_j\}$ , are randomly drawn from A with replacement so that i=j is allowed. he mean $\bar{x_{ij}}$ of the sample $\{x_i, x_j\}$ is defined as $\bar{x_{ij}}=\frac{1}{2}(x_i+x_j)}$ These sample means then form a set S = $\{\bar{x_{ij}}:i=1,2,...,n;j=1,2,,...,n\}$

a) Show that $\sum_{i=1}^{n}\sum_{j=1}^{n}(x_i+x_j)=2n\sum_{i=2} ^{n}x_i$

b) Show hat

i) the mean of all the elements of S is $\mu$

ii) the variance of all the elements of S is $\frac{1}{2}\sigma^2$

c) Suppose A = $\[x_i=i:i=1,2,...,10\}$
Find the mean and variance of all the elements of S.

d) Suppose A = $\{x_i=3i+4:i=1,2,...,10\}$
Using (c), find the mean and variance of all the elements of S.

solution

a)
$\sum_{i=1}^{n}\sum_{j=1}^{n}(x_i+x_j)=\sum_{i=1}^{ n}nx_i+n\sum_{j=1}^{n}x_j\\+\2n\sum_{i=1}^{n}x_i$

b)
i) mean of all the elements of S
$=E(\bar{x_{ij}})=\frac{1}{n^2}\sum_{i=1}^{n}\sum-{j=1}^{n}\bar{x_{ij}}$ (S has $n^2$ elements)

$=\frac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1} {2}(x_i+x_j)\\=\frac{1}{2n^2}(2n\sum_{i=1}^{n}x_i) \\=\frac{1}{2n^2}(2n)(n\mu)=\mu$

ii) Variance of all the elements of S
= Var(\bar{x_{ij}}= $\frac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}(\bar{x_{ ij}}-\mu)^2$ (S has $n^2$ elements)

$=\frac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}(\frac{x _i+x_j}{2}-\mu)^2\\=\frac{1}{4n^2}\sum_{i=1}^{n}\sum_{j=1}^{n }[(x_i-\mu)+(x_j-\mu)]^2\\=\frac{1}{4n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}[(x_i-\mu)^2+2(x_i-\mu)(x_j-\mu)+(x_j-\mu)^2]\\=\frac{1}{4n^2}[n\sum_{i=1}^{n}(x_i-\mu)^2+2\sum_{i=1}^{n}(x_i-\mu)\sum_{j=1}^{n}(x_j-\mu)+n\sum_{j=1}^{n}(x_j-\mu)^2]\\=\frac{1}{4n^2}[n(n\sigma^2)+2(0)(0)+n(n\sigma^2)]\\=\frac{1}{2}\sigma^2$

b) ii) alternative method

c) mean =5.5
variance = 8.25
$\thereforeE(\bar{x_{ij}})=5.5$ , Var( $\bar{x_{ij}}$ ) = $\frac{1}{2}(8.25)=4.125$

d) Using (c), mean = $3\times5.5+4=20.5$
Variance = $3^2\times8.25=74.25$

$\therefore$ $E(\bar{x_{ij}})=20.5$ . Var $(\bar{x_{ij}})$ = $\frac{1}{2}(74.25)=37.125$

Thread: Anybody know these kinds of statistics and their serious names? Do you know any books

LinkBack

Thread Tools

Display

Similar Math Help Forum Discussions

Buying two different kinds of toys.

Integrating different kinds of functions

Simplifying Expressions... Not sure which kinds

Statistics - Fiction/Non-fiction books

Search Tags