[Probability Theory] Final Review Notes: Basic Concepts of Mathematical Statistics

1. Population and sample

Overall : the whole of the research objects or the whole of a certain (or some) quantitative indicators of the research objects , use $X$ means (normal population:$X\text{~}N(\mu ,p^2)$）
个体：总体的每个元素
有限总体：含有有限个个体的总体
无限总体：含有无限个个体的总体
总体分布：数量指标$X$取不同值的比率（是客观存在的）

样本/子样：总体中取得的一部分个体
样本容量（$n$）：样本中所含个体的个数
抽样：取得样本的过程
抽样法：抽样过程所采取的方法
随机抽样法：每一个个体是从总体中随机抽取的
随机样本：采用随机抽样法得到的样本
$$\text{样本：}n\text{维随机向量}(X_1,X_2,\cdots ,X_n)\stackrel{\text{观测}}{⟶}\text{样本值：一组具体的实数}(x_1,x_2,\cdots ,x_n)$$简单随机样本：各$X_i$与$X$同分布且相互独立（不做特殊声明，样本均指简单随机样本）
简单随机抽样：获得简单随机样本的方法
简单随机样本$(X_1,X_2,\cdots ,X_n)$的分布函数：设总体$X$的分布函数为$F(x)$，则样本的分布函数为 F ( x 1 , x 2 , ⋯   , x n ) = P { X 1 ≤ x 1 , X 2 ≤ x 2 , ⋯   , X n ≤ x n } = ∏ i = 1 n P { X i ≤ x i } = ∏ i = 1 n F ( x i ) F(x_1,x_2,\cdots,x_n)=P\{X_1\le x_1,X_2\le x_2,\cdots,X_n\le x_n\}=\prod\limits_{i=1}^n P\{X_i\le x_i\}=\prod\limits_{i=1}^n F(x_i) F(x1​,x2​,⋯,xn​)=P{ X1​≤x1​,X2​≤x2​,⋯,Xn​≤xn​}=i=1∏n​P{ Xi​≤xi​}=i=1∏n​F(xi​)

• If overall $X$ is a continuous random variable (the probability density is$f\left(x\right)$ ), then samples$(X_1,X_2,\cdots ,X_n)$ is$f(x_1,x_2,\cdots ,x_n)=\prod _{i=1}^{n}f\left(x_i\right)$
• If overall $X$ is a discrete random variable (the distribution law isP { X = ai } = pi P\{X=a_i\}=p_iP{ X=ai​}=pi​), then the samples $(X_1,X_2,\cdots ,X_n)$的分布律为$P\{X_1=x_1,X_2=x_2,\cdots ,X_n=x_n\}=\prod _{i=1}^{n}P\{X=x_i\}$

2. Collation of sample data

1. Sample frequency distribution and frequency distribution

样本频数分布：样本值中不同数值在样本值中出现的频数（即次数）
样本频率分布：样本值中不同数值在样本值中出现的频率（即次数/样本容量）
设样本值中不同的数值记为$x_1^{\ast },x_2^{\ast },\cdots ,x_l^{\ast }$（递增），相应的频数为$m_1,m_2,\cdots ,m_l$（$\sum _{i=1}^l{m}_i=n$），则样本频数分布表：

指标$X$	$x_1^{\ast }$	$x_2^{\ast }$	$\cdots$	$x_l^{\ast }$
频数$m_i$	$m_1$	$m_2$	$\cdots$	$m_l$

样本频率分布表：

指标$X$	$x_1^{\ast }$	$x_2^{\ast }$	$\cdots$	$x_l^{\ast }$
频率$\frac{m_i}{n}$	$\frac{m_1}{n}$	$\frac{m_2}{n}$	$\cdots$	$\frac{m_l}{n}$

如果总体$X$ is a discrete random variable, then the event{ X = xi ∗ } \{X=x_i^*\}{ X=xi∗​} frequency$\frac{m_i}{n}$should be close to the probability pi p_i of its occurrence$p_i$.
If overall $X$ is a continuous random variable, then the event$\{X=x_i^{\ast }\}$ The probability of occurrence is$0$ , it is meaningless to examine the frequency distribution of the sample at this time, and it is necessary to examine the frequency histogram of the sample.

2. Frequency histogram

Let the overall $X$ is a continuous random variable with probability density$f(x)$，$(x_1,x_2,\cdots ,x_n)$ is from the overallXXA sample value of$X.$The method for making a frequency histogram is:

1. Organize data : put sample values $x_1,x_2,\cdots ,x_n$Sort from small to large $x_{(1)}\le x_{(2)}\le \cdots \le x_{(n)}$。

2. Grouping : in the interval [ a , b ] containing all observations [a,b]$[a,b]$中插入一些分点$a=t_0<t_1<\cdots <t_{l-1}<t_l=b$ grip$[a,b]$ divided into$l$个小区间：$$\underset{\underset{a}{\uparrow }}{t_0}{t}_1{t}_2\cdots t_{l-1}\underset{\underset{b}{\uparrow }}{t_l}$$Some concepts:

   • Group distance : the length between cells $d_i=t_i-t_{i-1}$
   • group median : the midpoint of the interval
   • Number of groups : the number of cells between $l$

   Generally, equal division is adopted (the distance between each group is equal), at this time $d_i=\frac{b-a}{l}$. Group distance $The\; choice\; of\; l$ :

   • $n>100$：$l$ take$10$ to$20$
   • $n\approx 50$：$l$ take$5$ or$6$

   Pay attention to the principle of division: make sure that there are sample observations falling into each interval.

3. Column grouping frequency distribution table : by $m_i$Indicates that the observed value falls into $(t_{i-1},t_i]$ (that is, the frequencyof this interval or this group),$f_i=\frac{m_i}{n}$For the frequency of this group , remember $y_i=\frac{f_i}{d_i}=\frac{m_i}{n{d}_i}$, to list the grouped data into a table:

group	group median	Frequency $m_i$	frequency $f_i$	$y_i$
$[27,30]$	$28.5$	$8$	$0.105$	$0.035$
$(30,33]$	$31.5$	$10$	$0.132$	$0.044$
$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$

4. Make a frequency histogram : at $On\; the\; xOy$ coordinate plane, usexxEach interval on the$x-$$(t_{i-1},t_i]$为底，以$y_i=\frac{f_i}{d_i}$Draw a row of vertical rectangles for the height to get the frequency histogram. Note that the height of the rectangle is $y_i=\frac{f_i}{d_i}$Instead of frequency $f_i$, is to be divided by the group distance, the purpose is to make the sum of the areas of all rectangles be $1$ . At this time the overall$X$ falls into the interval$(t_{i-1},t_i)$ probability$p_i\approx f_i$。
5. Make a probability density curve : smoothly connect the midpoints on the sides of each rectangle in the frequency histogram to obtain a curve, when $n$ and$When\; l$ is sufficiently large, this curve approximatesXXProbability density curve of$X$$y=f(x)$。

3. Empirical distribution function

Set sample values $(x_1,x_2,\cdots ,x_n)$ , its empirical distribution function is$$F_n(x)=\frac{1}{n}\sum _{i=1}^n\left[x_i\le x\right]$$其中$\left[x_i\le x\right]$ means when$x_i\le$Take 1 when$x$$1$，$x_i>$0when$x$$0$ . To sum up,$F_n\left(x\right)$ isnnLess than or equal to xx$among\; n$ sample values$x$ of$x_i$The number of divided by the sample size $n$ . In other words, it is less than or equal to$The\; ratio\; of\; the\; number\; of\; sample\; values\; of\; x$ to the total number of samples.

The empirical distribution function has the following properties:
(1) Monotonically increasing;
(2) Right continuous;
(3) $F_n(-\infty )=0$，$F_n(+\infty )=1$。

If the sample values ​​are given in a frequency distribution table, the empirical distribution function $F_n(x)$可具体表达为$$F_n(x)=\{\begin{array}{ll}0,& x<x_i^{\ast }\\ \frac{m_1+m_2+\cdots +m_i}{n},& x_i^{\ast }\le x<x_{i+1}^{\ast },(i=1,2,\cdots ,l-1)\\ 1,& x\ge x_l^{\ast }\end{array}$$Obviously $F_n\left(x\right)$ is a step function, at each$x_i^{\ast }$There is a jump here.

The empirical distribution function is not only related to the sample size, but also related to the obtained sample value $(x_1,x_2,\cdots ,x_n)$ related.

3. Statistics

1. The concept of statistics

Statistics : Let $(X_1,X_2,\cdots ,X_n)$ is from the overallXXA sample of$X$$T=g(X_1,X_2,\cdots ,X_n)$为$(X_1,X_2,\cdots ,X_n)$ of a real-valued function, and$g$ does not contain any unknown parameters, it is called$T$ is a sample$(X_1,X_2,\cdots ,X_n)$ is a statistic.
Observed value of statistics: if$(x_1,x_2,\cdots ,x_n)$ is the sample$(X_1,X_2,\cdots ,X_n)$ , then$t=g(x_1,x_2,\cdots ,x_n)$ is called the statisticTTAn observed value of$T.$

2. Several commonly used statistics

设$(X_1,X_2,\cdots ,X_n)$ is from the overallXXA sample of$X$$(x_1,x_2,\cdots ,x_n)$ is the observed value of this sample.

1) Sample mean

Sample mean : $\overline{X}=\frac{1}{n}\sum _{i=1}^n{X}_i$(Its observed value is recorded as $\overline{x}$）

设$E\left(X\right)=\mu$、$D(X)=p^2$ exists, then

• $E(\overline{X})=m$
• $D(\overline{X})=\frac{p^2}{n}$
• $(p)\underset{n\to \infty }{\lim }\overline{X}=\mu$ (For the definition of convergence according to probability, see[Probability Theory] Midterm Review Notes (Part 2): Law of Large Numbers and Central Limit Theorem)

2) Sample variance and sample standard deviation

样本方差：$S^2=\frac{1}{n-1}\sum _{i=1}^n(X_i-\overline{X}{)}^2=\frac{1}{n-1}\left(\sum _{i=1}^n{X}_i^2-n{\overline{X}}^2\right)$(the observed value is recorded as$s^2$ )
Sample standard deviation:$S=\sqrt{S^2}=\sqrt{\frac{1}{n-1}\sum _{i=1}^n(X_i-\overline{X}{)}^2}$(The observed value is denoted as $s$ )
They are quantities that reflect the degree of dispersion of the sample values.

设$E\left(X\right)=\mu$、$D(X)=p^2$ exists, then

• $E(S^2)=p^2$
• $(p)\underset{n\to \infty }{\lim }{S}^2=p^2$

3) Sample moments

Sample $k$阶原点矩：$A_k=\frac{1}{n}\sum _{i=1}^n{X}_i^k$(The observed value is recorded as $a_k$)
sample $K$ -order central moment:$B_k=\frac{1}{n}\sum _{i=1}^n(X_i-\overline{X}{)}^k$ (its observed value is denoted as$b_k$）

Obviously, $A_1=\overline{X}$，$B_1=0$，$B_2=\frac{n-1}{n}{S}^2$。

Let the overall $X$ 's$K$ -order origin moment$a_k=E\left(X^k\right)$exists, then

• $E(X^k)=a_k$
• $(p)\underset{n\to \infty }{\lim }{A}_k=a_k$

4) Order statistics

设$(X_1,X_2,\cdots ,X_n)$ is from the overallXXA sample of$X$$(x_1,x_2,\cdots ,x_n)$ is an observation for this sample. The observations$x_1,x_2,\cdots ,x_n$Arranged from small to large as $x_{(1)}\le x_{(2)}\le \cdots \le x_{(n)}$。

Define the statistic $X_{(k)}$The value is $x_{\left(k\right)}$（$k=1,2,\cdots ,n$ ), thus obtaining$n$ statistics$X_{(1)},X_{(2)},\cdots ,X_{(n)}$, and they satisfy $X_{\left(1\right)}\le X_{\left(2\right)}\le \cdots \le X_{\left(n\right)}$，称$X_{\left(1\right)},X_{\left(2\right)},\cdots ,X_{\left(n\right)}$is the order statistic or order statistic for this sample .

Minimum order statistic : X ( 1 ) = min ⁡ { X ( 1 ) , X ( 2 ) , ⋯ , X ( n ) } X_{(1)}=\min\{X_{(1)},X_{ (2)},\cdots,X_{(n)}\}X(1)​=min{ X(1)​,X(2)​,⋯,X(n)​}
Maximum order statistics:X ( n ) = max ⁡ { X ( 1 ) , X ( 2 ) , ⋯ , X ( n ) } X_{(n)}=\max\{X_{(1)},X_ {(2)},\cdots,X_{(n)}\}X(n)​=max{ X(1)​,X(2)​,⋯,X(n)​}

5) Extremely poor samples

Sample range : $R=X_{\left(n\right)}-X_{\left(1\right)}$(The observed value is recorded as $r=x_{\left(n\right)}-x_{\left(1\right)}$）

6) Sample $p$ -quantile

sample $p$ quantile: for$0<p<1$ , statistic$$M_p=\{\begin{array}{ll}{X}_{(\lceil np\rceil )},& np\text{is not an integer}\\ \frac{1}{2}\left(X_{(np)}+X_{(np+1)}\right),& np\text{is}\end{array}$$where $\lceil np\rceil$ stands for$np$ is rounded up, it is also equivalent to$np+1$ is rounded down.
Sample median:$p=\frac{1}{2}$The sample median when ( nnWhen$n$$X_{\left(\lceil \frac{n}{2}\rceil \right)}$，$n$为偶数时等于$\frac{1}{2}\left(X_{\left(\frac{n}{2}\right)}+X_{\left(\frac{n}{2}+1\right)}\right)$）

四、抽样分布

抽样分布：统计量的概率分布

1. $\Gamma$分布

$X$服从参数为$\alpha ,\lambda$的$\Gamma$分布：$X\text{~}\Gamma (\alpha ,\lambda )$，其中$\alpha >0,\lambda >0$

性质：

1. $X\text{~}\Gamma (\alpha ,\lambda )⟹E(X)=\frac{\alpha }{\lambda }$，$D(X)=\frac{\alpha }{{\lambda }^2}$
2. 设随机变量$X_1,X_2,\cdots ,X_m$相互独立，且$X_i\text{~}C(a_i,\lambda )$，则$\sum _{i=1}^m{X}_i\text{~}C\left(\sum _{i=1}^m{a}_i,l\right)$

2. $h^2$ distribution

In Γ \GammaTake α = n 2 \alpha=\frac{n}{2}in$the\; \Gamma \; distribution$$a=\frac{n}{2}$λ $l=\frac{1}{2}$，$The\; \Gamma$ distribution isnnχ 2 \chi^2of$n$$h^2$ distribution.
$Z$ obeysnnχ 2 \chi^2of$n$$h^{2Structure}$ :$Z\text{~}{h}^2(n)$

nature:

1. $Z\text{~}{h}^2\left(n\right)⟹E(Z)=n$，$D(Z)=2n$
2. If the random variable $Z_1,Z_2,\cdots ,Z_m$are independent of each other, and $Z_i\text{~}{h}^2(n_i)$，则$\sum _{i=1}^m{Z}_i\text{~}{\chi }^2\left(\sum _{i=1}^m{n}_i\right)$
3. 设随机变量$X_1,X_2,\cdots ,X_n$相互独立，且都服从标准正态分布$N(0,1)$，则随机变量${\chi }^2=\sum _{i=1}^n{X}_i^2$服从自由度为$n$的${\chi }^2$分布，即${\chi }^2\text{~}{\chi }^2(n)$。特别地，若$X\text{~}N(0,1)$，则$X^2\text{~}{\chi }^2(1)$。

3. $t$分布

$T$服从自由度为$n$的$t$分布：$T\text{~}t(n)$
$t$分布又称为学生氏分布。
$t$分布的概率密度关于$x=0$ symmetry ($\Gamma$ distribution,$h^2$ distribution,$The\; probability\; density\; of\; the\; F$ distribution is only at$x>0$ is positive), and$\underset{n\to \infty }{\lim }t(x;n)=\frac{1}{\sqrt{2p.m}}{e}^{-\frac{x^2}{2}}$, then $n\to \infty$ degrees of freedom are$n\text{'}tt$ _$The\; t$ distribution converges to the standard normal distribution$N(0,1)$。

性质：若$X\text{~}N(0,1)$，$Y\text{~}{\chi }^2(n)$，且$X$与$Y$相互独立，则$$T=\frac{X}{\sqrt{Y/n}}\text{~}t(n)$$

4. $F$分布

$F$服从自由度为$(n_1,n_2)$的$F$分布：$F\text{~}F(n_1,n_2)$

性质：

1. $X\text{~}{\chi }^2(n_1)$，$Y\text{~}{\chi }^2(n_2)$，且$X$与$Y$相互独立，则$$F=\frac{X/n_1}{Y/n_2}\text{~}F(n_1,n_2)$$
2. $F\text{~}F(n_1,n_2)⟹\frac{1}{F}\text{~}F(n_2,n_1)$ (just put XXin property 1$X$和YYIt can be proved by exchanging$Y\; )$

A detailed definition of the above distribution

1. $\Gamma$ distribution: if the random variable$X$ infinitesimal function$$f(x;a,l)=\{\begin{array}{ll}\frac{l^{}}{C\left(a\right)}{x}^{a-1}{e}^{-\lambda x},& x>0\\ 0,& x\le 0\end{array}$$where $a>0$，$l>0$ is a constant, it is called$X$ obeys parameters as$a,\lambda$的$\Gamma$ type, define$X\text{~}C(a,l)$ .
2. $h^2$ distribution: if the random variable$Z$ has probability density$$h^2(x;n)=\{\begin{array}{ll}\frac{1}{2^{\frac{n}{2}}C\left(\frac{n}{2}\right)}{x}^{\frac{n}{2}-1}{e}^{-\frac{x}{2}},& x>0\\ 0,& x\le 0\end{array}$$ZZ $Z$ obeysnnχ 2 \chi^2of$n$$h^2$ distribution, recorded as$Z\text{~}{h}^2(n)$。
3. $t$ distribution: if the random variable$T$具有概率密度$$t(x;n)=\frac{C\left(\frac{n+1}{2}\right)}{\sqrt{n\pi }C\left(\frac{n}{2}\right)}{\left(1+\frac{x^2}{n}\right)}^{-\frac{n+1}{2}}$$TT $T$ obeys$n\text{'}tt$ _$t$ distribution, recorded as$T\text{~}t(n)$。
4. $F$ distribution: if the random variable$F$具有概率密度$$f(x;n_1,n_2)=\{\begin{array}{l}\frac{C\left(\frac{n_1+n_2}{2}\right)}{C\left(\frac{n_1}{2}\right)C\left(\frac{n_2}{2}\right)}\left(\frac{n_1}{n_2}\right){\left(\frac{n_1}{n_2}x\right)}^{\frac{n_1}{2}-1}{\left(1+\frac{n_1}{n_2}x\right)}^{-\frac{n_1+n_2}{2}}\end{array}$$FF $F$ obeys degrees of freedom$(n_1,n_2)$ of$F$ distribution, recorded as$F\text{~}F(n_1,n_2)$。

quantile

Let random variable XXThe distribution function of$X$ is F ( x ) = P { X ≤ x } F(x)=P\{X\le x\}F(x)=P{ X≤x } .
lower side$p$ quantile: for$0<p<1$ , if$x_p$使 P { X ≤ x p } = F ( x p ) = p P\{X\le x_p\}=F(x_p)=p P{ X≤xp​}=F(xp​)=p , then it is called$x_p$is the distribution $F\left(x\right)$ (or random variable$X$ ) on the lower side$p$ -quantile.
upper side$\alpha$ quantile: for$0<a<1$ , if$x_{}$使$P\{X>x_{}\}=1-F\left(x_{}\right)=\alpha$ , then say$x_{}$is the distribution $F\left(x\right)$ (or random variable$X$ ) upper side$alpha$ quantile.

upper side $\alpha$ quantile = lower side$1-\alpha$ quantile;
lower side$p$ quantile = upper side$1-p$ -quantile.

Overall, the upper side $The\; \alpha$ quantile is such that$The\; probability\; that\; X$ is greater than it isα \alphaThe number of$\alpha \; .$

Standard normal distribution $N(0,1)$ on the upper side$\alpha$ quantile: for$u_{}$运动，$1-\Phi (u_{})=\alpha$ ;$u_{1-}=-u_{}$
$The\; upper\; side\; of\; the\; t\left(n\right)$ distribution$Alpha$ quantile: use$t_{}\left(n\right)$ means;$t_{1-}=-t_{}$
$h^2\left(n\right)$the upper side of the distribution$Alpha$ quantile: use$h_a^2\left(n\right)$ means
$F(n_1,n_2)$ distribution on the upper side$Alpha$ quantile: use$F_{}(n_1,n_2)$ display；$F_{}(n_1,n_2)=\frac{1}{F_{1-}(n_2,n_1)}$

If the probability density function of the distribution is about $x=0$ is symmetrical, then its upper side$1-The\; \alpha$ quantile is equal to the upper side$The\; opposite\; of\; the\; alpha$ quantile. Taking the standard normal distribution as an example, we know that$\Phi \left(u_{}\right)=1-\alpha$，$\Phi \left(u_{1-}\right)=\alpha$，则$\Phi (u_{})+\Phi \left(u_{1-}\right)=1$ Let$\Phi \left(u_{}\right)={\int }_{-\infty }^{u_{}}\phi \left(x\right)\mathrm{d}x={\int }_{-u_{}}^{+\infty }\phi \left(x\right)dx$ ,$\Phi \left(u_{1-}\right)={\int }_{-\infty }^{u_{1-}}\phi \left(x\right)dx$ , the sum of the two is$1$ , indicating that the lower limit of the former integral is equal to the upper limit of the latter integral, so$u_{1-}=-u_{}$. Similarly $t_{1-}=-t_{}$。

About $F_{}(n_1,n_2)=\frac{1}{F_{1-}(n_2,n_1)}$, the proof is as follows: Let $X\text{~}F(n_1,n_2)$，则P { X > F α ( n 1 , n 2 ) } = α P\{X>F_\alpha(n_1,n_2)\}=\alphaP{ X>Fa(n1​,n2​)}=α，$P\left\{\frac{1}{X}<\frac{1}{F_{}(n_1,n_2)}\right\}=\alpha$ ,而$\frac{1}{X}\text{~}F(n_2,n_1)$，故$P\left\{\frac{1}{X}<F_{1-}(n_2,n_1)\right\}=1-(1-a)=\alpha$，so1$\frac{1}{F_{}(n_1,n_2)}=F_{1-}(n_2,n_1)$。

Sampling distribution of a normal population

设$(X_1,X_2,\cdots ,X_n)$ is from the normal population$N(\mu ,p^2)$,$\overline{X}$is the sample mean, $S^2$ is the sample variance, then:

1. $$\overline{X}\text{~}N\left(m,\frac{p^2}{n}\right)$$
2. $$\frac{(n-1)S^2}{p^2}=\frac{\sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2}{p^2}\text{~}{h}^2(n-1)$$
3. $\overline{X}$with $S^2$ independent of each other
4. $$T=\frac{\sqrt{n}(\overline{X}-m}{S}\text{~}t(n-1)$$

设$(X_1,X_2,\cdots ,X_{n_1})$，$(Y_1,Y_2,\cdots ,Y_{n_2})$ are from$N(m_1,p^2)$，$N(m_2,p^2)$samples (note that the variances are equal), and the two samples are independent of each other,$\overline{X}=\frac{1}{n_1}\sum _{i=1}^{n_1}{X}_i$，$\overline{Y}=\frac{1}{n_2}\sum _{i=1}^{n_2}{Y}_i$，$S_{1n_1}^2=\frac{1}{n_1-1}\sum _{i=1}^{n_1}{\left(X_i-\overline{X}\right)}^2$，$S_{2n_2}^2=\frac{1}{n_2-1}\sum _{i=1}^{n_2}{\left(Y_i-\overline{Y}\right)}^2$ , then:

5. $$T=\frac{\left(\overline{X}-\overline{Y}\right)-(m_1-m_2)}{S_W\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\text{~}t(n_1+n_2-2)\left(S_W=\sqrt{\frac{(n_1-1)S_{1n_1}^2+(n_2-1)S_{2n_2}^2}{n_1+n_2-2}}\right)$$
6. $$F=\frac{p_2^2}{p_1^2}\frac{S_{1n_1}^2}{S_{2n_2}^2}\text{~}F(n_1-1,n_2-1)$$

explain:

1. 由$E(\overline{X})=\mu$、$D(\overline{X})=\frac{p^2}{n}$easy.
2. It needs complex linear algebra knowledge to prove, omitted.
3. omitted.
4. We know that $X\text{~}N(0,1)$ ,$Y\text{~}{h}^2\left(n\right)$，且$X,Y$ can be independently derived$T=\frac{X}{\sqrt{and/n}}\text{~}t\left(n\right)$ . Now we know that$\frac{\sqrt{n}(\overline{X}-m}{p}\text{~}N(0,1)$ ,$\frac{(n-1)S^2}{p^2}\text{~}{h}^2(n-1)$ , and both are independent, and$\frac{\sqrt{n}\left(\overline{X}-m\right)/}{\sqrt{(n-1)S^2/p^2/(n-1)}}=\frac{\sqrt{n}(\overline{X}-m}{S}$，故$\frac{\sqrt{n}(\overline{X}-m}{S}\text{~}t(n-1)$。
5. What you need to know is $\overline{X}-\overline{Y}\text{~}N\left(m_1-m_2,\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}{p}^2\right)$, so that$U=\frac{\left(\overline{X}-\overline{Y}\right)-(m_1-m_2)}{p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\text{~}N(0,1)$ N又$V=\frac{(n_1-1)S_{1n_1}^2}{p^2}+\frac{(n_2-1)S_{2n_2}^2}{p^2}\text{~}{h}^2(n_1+n_2-2)$，故$T=\frac{U}{\sqrt{V/(n_1+n_2-2)}}\text{~}t(n_1+n_2-2)$。
6. We know $\frac{(n_1-1)S_{1n_1}^2}{p_1^2}\text{~}{h}^2(n_1-1)$ ,$\frac{(n_2-1)S_{2n_2}^2}{p_2^2}\text{~}{h}^2(n_2-1)$ , and the two are independent of each other, so according to$F$ function function$F=\frac{\frac{(n_1-1)S_{1n_1}^2}{p_1^2}/(n_1-1)}{\frac{(n_2-1)S_{2n_2}^2}{p_2^2}/(n_2-1)}\text{~}F(n_1-1,n_2-1)$。