{
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   "source": [
    "# 机器学习数学基础知识\n",
    "更新地址：https://github.com/fengdu78/WZU-machine-learning-course\n",
    "\n",
    "整理编译：黄海广 haiguang2000@wzu.edu.cn\n",
    "\n",
    "数据科学需要一定的数学基础，但仅仅做应用的话，如果时间不多，不用学太深，了解基本公式即可，遇到问题再查吧。\n",
    "\n",
    "\n",
    "\n",
    "线性代数建议参考资料为：\n",
    "\n",
    "[1] KOLTER Z. Linear Algebra Review and Reference[J]. Available Online Http, 2015.\n",
    "\n",
    "[2] 同济大学数学系.线性代数[M]. 北京:人民邮电出版社,2017.\n",
    "\n",
    "重点推荐第一个资料，这是斯坦福大学人工智能方向的课程的数学复习资料，建议学习，其翻译版本见：https://zhuanlan.zhihu.com/p/466410564\n",
    "\n",
    "\n",
    "概率论与数理统计建议参考资料为：\n",
    "\n",
    "[1] ARIAN MALEKI, TOM DO. Review of Probability Theory[J]. Stanford University, 2019.\n",
    "\n",
    "[2] 同济大学数学系.概率论与数理统计[M]. 北京:人民邮电出版社,2017.\n",
    "\n",
    "重点推荐第一个资料，这是斯坦福大学人工智能方向的课程的数学复习资料，建议学习，其翻译版本见：https://zhuanlan.zhihu.com/p/466416090\n"
   ]
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   "id": "54fb2378",
   "metadata": {},
   "source": [
    "## 国内教材的数学知识整理\n",
    "以下是以前考研考博时候的数学笔记，难度应该在本科3年级左右。 "
   ]
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    "### 高等数学\n",
    "\n",
    "**1.导数定义：**\n",
    "\n",
    "导数和微分的概念\n",
    "\n",
    "$f'({{x}_{0}})=\\underset{\\Delta x\\to 0}{\\mathop{\\lim }}\\,\\frac{f({{x}_{0}}+\\Delta x)-f({{x}_{0}})}{\\Delta x}$    （1）\n",
    "\n",
    "或者：\n",
    "\n",
    "$f'({{x}_{0}})=\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,\\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}}$           （2）\n",
    "\n",
    "**2.左右导数导数的几何意义和物理意义**\n",
    "\n",
    "函数$f(x)$在$x_0$处的左、右导数分别定义为：\n",
    "\n",
    "左导数：${{{f}'}_{-}}({{x}_{0}})=\\underset{\\Delta x\\to {{0}^{-}}}{\\mathop{\\lim }}\\,\\frac{f({{x}_{0}}+\\Delta x)-f({{x}_{0}})}{\\Delta x}=\\underset{x\\to x_{0}^{-}}{\\mathop{\\lim }}\\,\\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}},(x={{x}_{0}}+\\Delta x)$\n",
    "\n",
    "右导数：${{{f}'}_{+}}({{x}_{0}})=\\underset{\\Delta x\\to {{0}^{+}}}{\\mathop{\\lim }}\\,\\frac{f({{x}_{0}}+\\Delta x)-f({{x}_{0}})}{\\Delta x}=\\underset{x\\to x_{0}^{+}}{\\mathop{\\lim }}\\,\\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}}$\n",
    "\n",
    "**3.函数的可导性与连续性之间的关系**\n",
    "\n",
    "**Th1:** 函数$f(x)$在$x_0$处可微$\\Leftrightarrow f(x)$在$x_0$处可导\n",
    "\n",
    "**Th2:** 若函数在点$x_0$处可导，则$y=f(x)$在点$x_0$处连续，反之则不成立。即函数连续不一定可导。\n",
    "\n",
    "**Th3:** ${f}'({{x}_{0}})$存在$\\Leftrightarrow {{{f}'}_{-}}({{x}_{0}})={{{f}'}_{+}}({{x}_{0}})$\n",
    "\n",
    "**4.平面曲线的切线和法线**\n",
    "\n",
    "切线方程 : $y-{{y}_{0}}=f'({{x}_{0}})(x-{{x}_{0}})$\n",
    "法线方程：$y-{{y}_{0}}=-\\frac{1}{f'({{x}_{0}})}(x-{{x}_{0}}),f'({{x}_{0}})\\ne 0$\n",
    "\n",
    "**5.四则运算法则**\n",
    "\n",
    "设函数$u=u(x)，v=v(x)$]在点$x$可导则\n",
    "\n",
    "(1) $(u\\pm v{)}'={u}'\\pm {v}'$       $d(u\\pm v)=du\\pm dv$\n",
    "\n",
    "(2)$(uv{)}'=u{v}'+v{u}'$        $d(uv)=udv+vdu$\n",
    "\n",
    "(3) $(\\frac{u}{v}{)}'=\\frac{v{u}'-u{v}'}{{{v}^{2}}}(v\\ne 0)$       $d(\\frac{u}{v})=\\frac{vdu-udv}{{{v}^{2}}}$\n",
    "\n",
    "**6.基本导数与微分表**\n",
    "\n",
    "(1) $y=c$（常数）       ${y}'=0$          $dy=0$\n",
    "\n",
    "(2) $y={{x}^{\\alpha }}$($\\alpha $为实数)    ${y}'=\\alpha {{x}^{\\alpha -1}}$      $dy=\\alpha {{x}^{\\alpha -1}}dx$\n",
    "\n",
    "(3) $y={{a}^{x}}$      ${y}'={{a}^{x}}\\ln a$         $dy={{a}^{x}}\\ln adx$\n",
    "  特例:   $({{{e}}^{x}}{)}'={{{e}}^{x}}$             $d({{{e}}^{x}})={{{e}}^{x}}dx$\n",
    "\n",
    "(4) $y={{\\log }_{a}}x$   ${y}'=\\frac{1}{x\\ln a}$           \n",
    "\n",
    "$dy=\\frac{1}{x\\ln a}dx$\n",
    "  特例:$y=\\ln x$                      $(\\ln x{)}'=\\frac{1}{x}$       $d(\\ln x)=\\frac{1}{x}dx$\n",
    "\n",
    "(5) $y=\\sin x$         \n",
    "\n",
    "${y}'=\\cos x$        $d(\\sin x)=\\cos xdx$\n",
    "\n",
    "(6) $y=\\cos x$      \n",
    "\n",
    "${y}'=-\\sin x$       $d(\\cos x)=-\\sin xdx$\n",
    "\n",
    "(7) $y=\\tan x$  \n",
    "\n",
    "${y}'=\\frac{1}{{{\\cos }^{2}}x}={{\\sec }^{2}}x$  $d(\\tan x)={{\\sec }^{2}}xdx$\n",
    "(8) $y=\\cot x$ ${y}'=-\\frac{1}{{{\\sin }^{2}}x}=-{{\\csc }^{2}}x$  $d(\\cot x)=-{{\\csc }^{2}}xdx$\n",
    "(9) $y=\\sec x$ ${y}'=\\sec x\\tan x$     \n",
    "\n",
    " $d(\\sec x)=\\sec x\\tan xdx$\n",
    "(10) $y=\\csc x$ ${y}'=-\\csc x\\cot x$    \n",
    "\n",
    "$d(\\csc x)=-\\csc x\\cot xdx$\n",
    "(11) $y=\\arcsin x$  \n",
    "\n",
    "${y}'=\\frac{1}{\\sqrt{1-{{x}^{2}}}}$   \n",
    "\n",
    "$d(\\arcsin x)=\\frac{1}{\\sqrt{1-{{x}^{2}}}}dx$\n",
    "\n",
    "(12) $y=\\arccos x$ \n",
    "\n",
    "${y}'=-\\frac{1}{\\sqrt{1-{{x}^{2}}}}$     $d(\\arccos x)=-\\frac{1}{\\sqrt{1-{{x}^{2}}}}dx$\n",
    "\n",
    "(13) $y=\\arctan x$ \n",
    "\n",
    "${y}'=\\frac{1}{1+{{x}^{2}}}$     $d(\\arctan x)=\\frac{1}{1+{{x}^{2}}}dx$\n",
    "\n",
    "(14) $y=\\operatorname{arc}\\cot x$      \n",
    "\n",
    "${y}'=-\\frac{1}{1+{{x}^{2}}}$   \n",
    "\n",
    "$d(\\operatorname{arc}\\cot x)=-\\frac{1}{1+{{x}^{2}}}dx$\n",
    "\n",
    "(15) $y=shx$    \n",
    "\n",
    "${y}'=chx$       $d(shx)=chxdx$\n",
    "\n",
    "(16) $y=chx$    \n",
    "\n",
    "${y}'=shx$       $d(chx)=shxdx$\n",
    "\n",
    "**7.复合函数，反函数，隐函数以及参数方程所确定的函数的微分法**\n",
    "\n",
    "(1) 反函数的运算法则: 设$y=f(x)$在点$x$的某邻域内单调连续，在点$x$处可导且${f}'(x)\\ne 0$，则其反函数在点$x$所对应的$y$处可导，并且有$\\frac{dy}{dx}=\\frac{1}{\\frac{dx}{dy}}$\n",
    "\n",
    "(2) 复合函数的运算法则:若$\\mu =\\varphi (x)$在点$x$可导,而$y=f(\\mu )$在对应点$\\mu $($\\mu =\\varphi (x)$)可导,则复合函数$y=f(\\varphi (x))$在点$x$可导,且${y}'={f}'(\\mu )\\cdot {\\varphi }'(x)$\n",
    "\n",
    "(3) 隐函数导数$\\frac{dy}{dx}$的求法一般有三种方法：\n",
    "\n",
    "1)方程两边对$x$求导，要记住$y$是$x$的函数，则$y$的函数是$x$的复合函数.例如$\\frac{1}{y}$，${{y}^{2}}$，$ln y$，${{{e}}^{y}}$等均是$x$的复合函数.\n",
    "\n",
    "对$x$求导应按复合函数连锁法则做.\n",
    "\n",
    "2)公式法.由$F(x,y)=0$知 $\\frac{dy}{dx}=-\\frac{{{{{F}'}}_{x}}(x,y)}{{{{{F}'}}_{y}}(x,y)}$,其中，${{{F}'}_{x}}(x,y)$，\n",
    "${{{F}'}_{y}}(x,y)$分别表示$F(x,y)$对$x$和$y$的偏导数\n",
    "\n",
    "3)利用微分形式不变性\n",
    "\n",
    "**8.常用高阶导数公式**\n",
    "\n",
    "（1）$({{a}^{x}}){{\\,}^{(n)}}={{a}^{x}}{{\\ln }^{n}}a\\quad (a>{0})\\quad \\quad ({{{e}}^{x}}){{\\,}^{(n)}}={e}{{\\,}^{x}}$\n",
    "\n",
    "（2）$(\\sin kx{)}{{\\,}^{(n)}}={{k}^{n}}\\sin (kx+n\\cdot \\frac{\\pi }{{2}})$\n",
    "\n",
    "（3）$(\\cos kx{)}{{\\,}^{(n)}}={{k}^{n}}\\cos (kx+n\\cdot \\frac{\\pi }{{2}})$\n",
    "\n",
    "（4）$({{x}^{m}}){{\\,}^{(n)}}=m(m-1)\\cdots (m-n+1){{x}^{m-n}}$\n",
    "\n",
    "（5）$(\\ln x){{\\,}^{(n)}}={{(-{1})}^{(n-{1})}}\\frac{(n-{1})!}{{{x}^{n}}}$\n",
    "\n",
    "（6）莱布尼兹公式：若$u(x)\\,,v(x)$均$n$阶可导，则\n",
    " ${{(uv)}^{(n)}}=\\sum\\limits_{i={0}}^{n}{c_{n}^{i}{{u}^{(i)}}{{v}^{(n-i)}}}$，其中${{u}^{({0})}}=u$，${{v}^{({0})}}=v$\n",
    "\n",
    "**9.微分中值定理，泰勒公式**\n",
    "\n",
    "**Th1:**(费马定理)\n",
    "\n",
    "若函数$f(x)$满足条件：\n",
    "\n",
    "(1)函数$f(x)$在${{x}_{0}}$的某邻域内有定义，并且在此邻域内恒有\n",
    "$f(x)\\le f({{x}_{0}})$或$f(x)\\ge f({{x}_{0}})$,\n",
    "\n",
    "(2) $f(x)$在${{x}_{0}}$处可导,则有 ${f}'({{x}_{0}})=0$\n",
    "\n",
    "**Th2:**(罗尔定理) \n",
    "\n",
    "设函数$f(x)$满足条件：\n",
    "\n",
    "(1)在闭区间$[a,b]$上连续；\n",
    "\n",
    "(2)在$(a,b)$内可导；\n",
    "\n",
    "(3)$f(a)=f(b)$；\n",
    "\n",
    "则在$(a,b)$内一存在个$\\xi $，使  ${f}'(\\xi )=0$\n",
    "\n",
    "**Th3:** (拉格朗日中值定理) \n",
    "\n",
    "设函数$f(x)$满足条件：\n",
    "(1)在$[a,b]$上连续；\n",
    "\n",
    "(2)在$(a,b)$内可导；\n",
    "\n",
    "则在$(a,b)$内一存在个$\\xi $，使  $\\frac{f(b)-f(a)}{b-a}={f}'(\\xi )$\n",
    "\n",
    "**Th4:** (柯西中值定理)\n",
    "\n",
    " 设函数$f(x)$，$g(x)$满足条件：\n",
    " \n",
    "(1) 在$[a,b]$上连续；\n",
    "\n",
    "(2) 在$(a,b)$内可导且${f}'(x)$，${g}'(x)$均存在，且${g}'(x)\\ne 0$\n",
    "\n",
    "则在$(a,b)$内存在一个$\\xi $，使  $\\frac{f(b)-f(a)}{g(b)-g(a)}=\\frac{{f}'(\\xi )}{{g}'(\\xi )}$\n",
    "\n",
    "**10.洛必达法则**\n",
    "\n",
    "法则Ⅰ ($\\frac{0}{0}$型)\n",
    "\n",
    "设函数$f\\left( x \\right),g\\left( x \\right)$满足条件：\n",
    "\n",
    " $\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,f\\left( x \\right)=0,\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,g\\left( x \\right)=0$; \n",
    "\n",
    "$f\\left( x \\right),g\\left( x \\right)$在${{x}_{0}}$的邻域内可导，(在${{x}_{0}}$处可除外)且${g}'\\left( x \\right)\\ne 0$;\n",
    "\n",
    "$\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,\\frac{{f}'\\left( x \\right)}{{g}'\\left( x \\right)}$存在(或$\\infty $)。\n",
    "\n",
    "则:\n",
    "$\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,\\frac{f\\left( x \\right)}{g\\left( x \\right)}=\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,\\frac{{f}'\\left( x \\right)}{{g}'\\left( x \\right)}$。\n",
    "法则${{I}'}$ ($\\frac{0}{0}$型)设函数$f\\left( x \\right),g\\left( x \\right)$满足条件：\n",
    "$\\underset{x\\to \\infty }{\\mathop{\\lim }}\\,f\\left( x \\right)=0,\\underset{x\\to \\infty }{\\mathop{\\lim }}\\,g\\left( x \\right)=0$;\n",
    "\n",
    "存在一个$X>0$,当$\\left| x \\right|>X$时,$f\\left( x \\right),g\\left( x \\right)$可导,且${g}'\\left( x \\right)\\ne 0$;$\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,\\frac{{f}'\\left( x \\right)}{{g}'\\left( x \\right)}$存在(或$\\infty $)。\n",
    "\n",
    "则:\n",
    "$\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,\\frac{f\\left( x \\right)}{g\\left( x \\right)}=\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,\\frac{{f}'\\left( x \\right)}{{g}'\\left( x \\right)}$\n",
    "法则Ⅱ($\\frac{\\infty }{\\infty }$型) 设函数$f\\left( x \\right),g\\left( x \\right)$\n",
    "\n",
    "满足条件：\n",
    "$\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,f\\left( x \\right)=\\infty ,\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,g\\left( x \\right)=\\infty $;   $f\\left( x \\right),g\\left( x \\right)$在${{x}_{0}}$ 的邻域内可导(在${{x}_{0}}$处可除外)且${g}'\\left( x \\right)\\ne 0$;$\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,\\frac{{f}'\\left( x \\right)}{{g}'\\left( x \\right)}$存在(或$\\infty $)。则\n",
    "$\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,\\frac{f\\left( x \\right)}{g\\left( x \\right)}=\\underset{x\\to {{x}_{0}}}{\\mathop{\\lim }}\\,\\frac{{f}'\\left( x \\right)}{{g}'\\left( x \\right)}.$同理法则${I{I}'}$($\\frac{\\infty }{\\infty }$型)仿法则${{I}'}$可写出。\n",
    "\n",
    "**11.泰勒公式**\n",
    "\n",
    "设函数$f(x)$在点${{x}_{0}}$处的某邻域内具有$n+1$阶导数，则对该邻域内异于${{x}_{0}}$的任意点$x$，在${{x}_{0}}$与$x$之间至少存在\n",
    "一个$\\xi $，使得：\n",
    "\n",
    "$f(x)=f({{x}_{0}})+{f}'({{x}_{0}})(x-{{x}_{0}})+\\frac{1}{2!}{f}''({{x}_{0}}){{(x-{{x}_{0}})}^{2}}+\\cdots $ \n",
    "$+\\frac{{{f}^{(n)}}({{x}_{0}})}{n!}{{(x-{{x}_{0}})}^{n}}+{{R}_{n}}(x)$\n",
    " 其中 ${{R}_{n}}(x)=\\frac{{{f}^{(n+1)}}(\\xi )}{(n+1)!}{{(x-{{x}_{0}})}^{n+1}}$称为$f(x)$在点${{x}_{0}}$处的$n$阶泰勒余项。\n",
    "\n",
    "令${{x}_{0}}=0$，则$n$阶泰勒公式\n",
    "$f(x)=f(0)+{f}'(0)x+\\frac{1}{2!}{f}''(0){{x}^{2}}+\\cdots +\\frac{{{f}^{(n)}}(0)}{n!}{{x}^{n}}+{{R}_{n}}(x)$……(1)\n",
    "其中 ${{R}_{n}}(x)=\\frac{{{f}^{(n+1)}}(\\xi )}{(n+1)!}{{x}^{n+1}}$，$\\xi $在0与$x$之间.(1)式称为麦克劳林公式\n",
    "\n",
    "**常用五种函数在${{x}_{0}}=0$处的泰勒公式**\n",
    "\n",
    "(1) ${{{e}}^{x}}=1+x+\\frac{1}{2!}{{x}^{2}}+\\cdots +\\frac{1}{n!}{{x}^{n}}+\\frac{{{x}^{n+1}}}{(n+1)!}{{e}^{\\xi }}$ \n",
    "\n",
    "或 $=1+x+\\frac{1}{2!}{{x}^{2}}+\\cdots +\\frac{1}{n!}{{x}^{n}}+o({{x}^{n}})$\n",
    "\n",
    "(2) $\\sin x=x-\\frac{1}{3!}{{x}^{3}}+\\cdots +\\frac{{{x}^{n}}}{n!}\\sin \\frac{n\\pi }{2}+\\frac{{{x}^{n+1}}}{(n+1)!}\\sin (\\xi +\\frac{n+1}{2}\\pi )$\n",
    "\n",
    "或  $=x-\\frac{1}{3!}{{x}^{3}}+\\cdots +\\frac{{{x}^{n}}}{n!}\\sin \\frac{n\\pi }{2}+o({{x}^{n}})$\n",
    "\n",
    "(3) $\\cos x=1-\\frac{1}{2!}{{x}^{2}}+\\cdots +\\frac{{{x}^{n}}}{n!}\\cos \\frac{n\\pi }{2}+\\frac{{{x}^{n+1}}}{(n+1)!}\\cos (\\xi +\\frac{n+1}{2}\\pi )$\n",
    "\n",
    "或   $=1-\\frac{1}{2!}{{x}^{2}}+\\cdots +\\frac{{{x}^{n}}}{n!}\\cos \\frac{n\\pi }{2}+o({{x}^{n}})$\n",
    "\n",
    "(4) $\\ln (1+x)=x-\\frac{1}{2}{{x}^{2}}+\\frac{1}{3}{{x}^{3}}-\\cdots +{{(-1)}^{n-1}}\\frac{{{x}^{n}}}{n}+\\frac{{{(-1)}^{n}}{{x}^{n+1}}}{(n+1){{(1+\\xi )}^{n+1}}}$\n",
    "\n",
    "或      $=x-\\frac{1}{2}{{x}^{2}}+\\frac{1}{3}{{x}^{3}}-\\cdots +{{(-1)}^{n-1}}\\frac{{{x}^{n}}}{n}+o({{x}^{n}})$\n",
    "\n",
    "(5) ${{(1+x)}^{m}}=1+mx+\\frac{m(m-1)}{2!}{{x}^{2}}+\\cdots +\\frac{m(m-1)\\cdots (m-n+1)}{n!}{{x}^{n}}$ \n",
    "$+\\frac{m(m-1)\\cdots (m-n+1)}{(n+1)!}{{x}^{n+1}}{{(1+\\xi )}^{m-n-1}}$ \n",
    "\n",
    "或 ${{(1+x)}^{m}}=1+mx+\\frac{m(m-1)}{2!}{{x}^{2}}+\\cdots $ $+\\frac{m(m-1)\\cdots (m-n+1)}{n!}{{x}^{n}}+o({{x}^{n}})$\n",
    "\n",
    "**12.函数单调性的判断**\n",
    "\n",
    "**Th1:**  设函数$f(x)$在$(a,b)$区间内可导，如果对$\\forall x\\in (a,b)$，都有$f\\,'(x)>0$（或$f\\,'(x)<0$），则函数$f(x)$在$(a,b)$内是单调增加的（或单调减少）\n",
    "\n",
    "**Th2:** （取极值的必要条件）设函数$f(x)$在${{x}_{0}}$处可导，且在${{x}_{0}}$处取极值，则$f\\,'({{x}_{0}})=0$。\n",
    "\n",
    "**Th3:** （取极值的第一充分条件）设函数$f(x)$在${{x}_{0}}$的某一邻域内可微，且$f\\,'({{x}_{0}})=0$（或$f(x)$在${{x}_{0}}$处连续，但$f\\,'({{x}_{0}})$不存在。）\n",
    "\n",
    "(1)若当$x$经过${{x}_{0}}$时，$f\\,'(x)$由“+”变“-”，则$f({{x}_{0}})$为极大值；\n",
    "\n",
    "(2)若当$x$经过${{x}_{0}}$时，$f\\,'(x)$由“-”变“+”，则$f({{x}_{0}})$为极小值；\n",
    "\n",
    "(3)若$f\\,'(x)$经过$x={{x}_{0}}$的两侧不变号，则$f({{x}_{0}})$不是极值。\n",
    "\n",
    "**Th4:** (取极值的第二充分条件)设$f(x)$在点${{x}_{0}}$处有$f''(x)\\ne 0$，且$f\\,'({{x}_{0}})=0$，则 当$f'\\,'({{x}_{0}})<0$时，\n",
    "$f({{x}_{0}})$为极大值；\n",
    "\n",
    "当$f'\\,'({{x}_{0}})>0$时，$f({{x}_{0}})$为极小值。\n",
    "\n",
    "注：如果$f'\\,'({{x}_{0}})<0$，此方法失效。\n",
    "\n",
    "**13.渐近线的求法**\n",
    "(1)水平渐近线   若$\\underset{x\\to +\\infty }{\\mathop{\\lim }}\\,f(x)=b$，或$\\underset{x\\to -\\infty }{\\mathop{\\lim }}\\,f(x)=b$，则\n",
    "\n",
    "$y=b$称为函数$y=f(x)$的水平渐近线。\n",
    "\n",
    "(2)铅直渐近线   若$\\underset{x\\to x_{0}^{-}}{\\mathop{\\lim }}\\,f(x)=\\infty $，或$\\underset{x\\to x_{0}^{+}}{\\mathop{\\lim }}\\,f(x)=\\infty $，则\n",
    "\n",
    "$x={{x}_{0}}$称为$y=f(x)$的铅直渐近线。\n",
    "\n",
    "(3)斜渐近线   若$a=\\underset{x\\to \\infty }{\\mathop{\\lim }}\\,\\frac{f(x)}{x},\\quad b=\\underset{x\\to \\infty }{\\mathop{\\lim }}\\,[f(x)-ax]$，则\n",
    "$y=ax+b$称为$y=f(x)$的斜渐近线。\n",
    "\n",
    "**14.函数凹凸性的判断**\n",
    "\n",
    "**Th1:** (凹凸性的判别定理）若在I上$f''(x)<0$（或$f''(x)>0$），则$f(x)$在I上是凸的（或凹的）。\n",
    "\n",
    "**Th2:** (拐点的判别定理1)若在${{x}_{0}}$处$f''(x)=0$，（或$f''(x)$不存在），当$x$变动经过${{x}_{0}}$时，$f''(x)$变号，则$({{x}_{0}},f({{x}_{0}}))$为拐点。\n",
    "\n",
    "**Th3:** (拐点的判别定理2)设$f(x)$在${{x}_{0}}$点的某邻域内有三阶导数，且$f''(x)=0$，$f'''(x)\\ne 0$，则$({{x}_{0}},f({{x}_{0}}))$为拐点。\n",
    "\n",
    "**15.弧微分**\n",
    "\n",
    "$dS=\\sqrt{1+y{{'}^{2}}}dx$\n",
    "\n",
    "**16.曲率**\n",
    "\n",
    "曲线$y=f(x)$在点$(x,y)$处的曲率$k=\\frac{\\left| y'' \\right|}{{{(1+y{{'}^{2}})}^{\\tfrac{3}{2}}}}$。\n",
    "对于参数方程$\\left\\{ \\begin{align}  & x=\\varphi (t) \\\\  & y=\\psi (t) \\\\ \\end{align} \\right.,$$k=\\frac{\\left| \\varphi '(t)\\psi ''(t)-\\varphi ''(t)\\psi '(t) \\right|}{{{[\\varphi {{'}^{2}}(t)+\\psi {{'}^{2}}(t)]}^{\\tfrac{3}{2}}}}$。\n",
    "\n",
    "**17.曲率半径**\n",
    "\n",
    "曲线在点$M$处的曲率$k(k\\ne 0)$与曲线在点$M$处的曲率半径$\\rho $有如下关系：$\\rho =\\frac{1}{k}$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89502fd9",
   "metadata": {},
   "source": [
    "### 线性代数\n",
    "\n",
    "#### 行列式\n",
    "\n",
    "**1.行列式按行（列）展开定理**\n",
    "\n",
    "(1) 设$A = ( a_{{ij}} )_{n \\times n}$，则：$a_{i1}A_{j1} +a_{i2}A_{j2} + \\cdots + a_{{in}}A_{{jn}} = \\begin{cases}|A|,i=j\\\\ 0,i \\neq j\\end{cases}$\n",
    "\n",
    "\n",
    "或$a_{1i}A_{1j} + a_{2i}A_{2j} + \\cdots + a_{{ni}}A_{{nj}} = \\begin{cases}|A|,i=j\\\\ 0,i \\neq j\\end{cases}$即 $AA^{*} = A^{*}A = \\left| A \\right|E,$其中：$A^{*} = \\begin{pmatrix} A_{11} & A_{12} & \\ldots & A_{1n} \\\\ A_{21} & A_{22} & \\ldots & A_{2n} \\\\ \\ldots & \\ldots & \\ldots & \\ldots \\\\ A_{n1} & A_{n2} & \\ldots & A_{{nn}} \\\\ \\end{pmatrix} = (A_{{ji}}) = {(A_{{ij}})}^{T}$\n",
    "\n",
    "$D_{n} = \\begin{vmatrix} 1 & 1 & \\ldots & 1 \\\\ x_{1} & x_{2} & \\ldots & x_{n} \\\\ \\ldots & \\ldots & \\ldots & \\ldots \\\\ x_{1}^{n - 1} & x_{2}^{n - 1} & \\ldots & x_{n}^{n - 1} \\\\ \\end{vmatrix} = \\prod_{1 \\leq j < i \\leq n}^{}\\,(x_{i} - x_{j})$\n",
    "\n",
    "(2) 设$A,B$为$n$阶方阵，则$\\left| {AB} \\right| = \\left| A \\right|\\left| B \\right| = \\left| B \\right|\\left| A \\right| = \\left| {BA} \\right|$，但$\\left| A \\pm B \\right| = \\left| A \\right| \\pm \\left| B \\right|$不一定成立。\n",
    "\n",
    "(3) $\\left| {kA} \\right| = k^{n}\\left| A \\right|$,$A$为$n$阶方阵。\n",
    "\n",
    "(4) 设$A$为$n$阶方阵，$|A^{T}| = |A|;|A^{- 1}| = |A|^{- 1}$（若$A$可逆），$|A^{*}| = |A|^{n - 1}$\n",
    "\n",
    "$n \\geq 2$\n",
    "\n",
    "(5) $\\left| \\begin{matrix}  & {A\\quad O} \\\\  & {O\\quad B} \\\\ \\end{matrix} \\right| = \\left| \\begin{matrix}  & {A\\quad C} \\\\  & {O\\quad B} \\\\ \\end{matrix} \\right| = \\left| \\begin{matrix}  & {A\\quad O} \\\\  & {C\\quad B} \\\\ \\end{matrix} \\right| =| A||B|$\n",
    "，$A,B$为方阵，但$\\left| \\begin{matrix} {O} & A_{m \\times m} \\\\  B_{n \\times n} & { O} \\\\ \\end{matrix} \\right| = ({- 1)}^{{mn}}|A||B|$ 。\n",
    "\n",
    "(6) 范德蒙行列式$D_{n} = \\begin{vmatrix} 1 & 1 & \\ldots & 1 \\\\ x_{1} & x_{2} & \\ldots & x_{n} \\\\ \\ldots & \\ldots & \\ldots & \\ldots \\\\ x_{1}^{n - 1} & x_{2}^{n 1} & \\ldots & x_{n}^{n - 1} \\\\ \\end{vmatrix} =  \\prod_{1 \\leq j < i \\leq n}^{}\\,(x_{i} - x_{j})$\n",
    "\n",
    "设$A$是$n$阶方阵，$\\lambda_{i}(i = 1,2\\cdots,n)$是$A$的$n$个特征值，则\n",
    "$|A| = \\prod_{i = 1}^{n}\\lambda_{i}$\n",
    "\n",
    "#### 矩阵\n",
    "\n",
    "矩阵：$m \\times n$个数$a_{{ij}}$排成$m$行$n$列的表格$\\begin{bmatrix}  a_{11}\\quad a_{12}\\quad\\cdots\\quad a_{1n} \\\\ a_{21}\\quad a_{22}\\quad\\cdots\\quad a_{2n} \\\\ \\quad\\cdots\\cdots\\cdots\\cdots\\cdots \\\\  a_{m1}\\quad a_{m2}\\quad\\cdots\\quad a_{{mn}} \\\\ \\end{bmatrix}$ 称为矩阵，简记为$A$，或者$\\left( a_{{ij}} \\right)_{m \\times n}$ 。若$m = n$，则称$A$是$n$阶矩阵或$n$阶方阵。\n",
    "\n",
    "**矩阵的线性运算**\n",
    "\n",
    "**1.矩阵的加法**\n",
    "\n",
    "设$A = (a_{{ij}}),B = (b_{{ij}})$是两个$m \\times n$矩阵，则$m \\times n$ 矩阵$C = c_{{ij}}) = a_{{ij}} + b_{{ij}}$称为矩阵$A$与$B$的和，记为$A + B = C$ 。\n",
    "\n",
    "**2.矩阵的数乘**\n",
    "\n",
    "设$A = (a_{{ij}})$是$m \\times n$矩阵，$k$是一个常数，则$m \\times n$矩阵$(ka_{{ij}})$称为数$k$与矩阵$A$的数乘，记为${kA}$。\n",
    "\n",
    "**3.矩阵的乘法**\n",
    "\n",
    "设$A = (a_{{ij}})$是$m \\times n$矩阵，$B = (b_{{ij}})$是$n \\times s$矩阵，那么$m \\times s$矩阵$C = (c_{{ij}})$，其中$c_{{ij}} = a_{i1}b_{1j} + a_{i2}b_{2j} + \\cdots + a_{{in}}b_{{nj}} = \\sum_{k =1}^{n}{a_{{ik}}b_{{kj}}}$称为${AB}$的乘积，记为$C = AB$ 。\n",
    "\n",
    "**4.** $\\mathbf{A}^{\\mathbf{T}}$**、**$\\mathbf{A}^{\\mathbf{-1}}$**、**$\\mathbf{A}^{\\mathbf{*}}$**三者之间的关系**\n",
    "\n",
    "(1) ${(A^{T})}^{T} = A,{(AB)}^{T} = B^{T}A^{T},{(kA)}^{T} = kA^{T},{(A \\pm B)}^{T} = A^{T} \\pm B^{T}$\n",
    "\n",
    "(2) $\\left( A^{- 1} \\right)^{- 1} = A,\\left( {AB} \\right)^{- 1} = B^{- 1}A^{- 1},\\left( {kA} \\right)^{- 1} = \\frac{1}{k}A^{- 1},$\n",
    "\n",
    "但 ${(A \\pm B)}^{- 1} = A^{- 1} \\pm B^{- 1}$不一定成立。\n",
    "\n",
    "(3) $\\left( A^{*} \\right)^{*} = |A|^{n - 2}\\ A\\ \\ (n \\geq 3)$，$\\left({AB} \\right)^{*} = B^{*}A^{*},$ $\\left( {kA} \\right)^{*} = k^{n -1}A^{*}{\\ \\ }\\left( n \\geq 2 \\right)$\n",
    "\n",
    "但$\\left( A \\pm B \\right)^{*} = A^{*} \\pm B^{*}$不一定成立。\n",
    "\n",
    "(4) ${(A^{- 1})}^{T} = {(A^{T})}^{- 1},\\ \\left( A^{- 1} \\right)^{*} ={(AA^{*})}^{- 1},{(A^{*})}^{T} = \\left( A^{T} \\right)^{*}$\n",
    "\n",
    "**5.有关**$\\mathbf{A}^{\\mathbf{*}}$**的结论**\n",
    "\n",
    "(1) $AA^{*} = A^{*}A = |A|E$\n",
    "\n",
    "(2) $|A^{*}| = |A|^{n - 1}\\ (n \\geq 2),\\ \\ \\ \\ {(kA)}^{*} = k^{n -1}A^{*},{{\\ \\ }\\left( A^{*} \\right)}^{*} = |A|^{n - 2}A(n \\geq 3)$\n",
    "\n",
    "(3) 若$A$可逆，则$A^{*} = |A|A^{- 1},{(A^{*})}^{*} = \\frac{1}{|A|}A$\n",
    "\n",
    "(4) 若$A$为$n$阶方阵，则：\n",
    "\n",
    "$r(A^*)=\\begin{cases}n,\\quad r(A)=n\\\\ 1,\\quad r(A)=n-1\\\\ 0,\\quad r(A)<n-1\\end{cases}$\n",
    "\n",
    "**6.有关**$\\mathbf{A}^{\\mathbf{- 1}}$**的结论**\n",
    "\n",
    "$A$可逆$\\Leftrightarrow AB = E; \\Leftrightarrow |A| \\neq 0; \\Leftrightarrow r(A) = n;$\n",
    "\n",
    "$\\Leftrightarrow A$可以表示为初等矩阵的乘积；$\\Leftrightarrow A;\\Leftrightarrow Ax = 0$。\n",
    "\n",
    "**7.有关矩阵秩的结论**\n",
    "\n",
    "(1) 秩$r(A)$=行秩=列秩；\n",
    "\n",
    "(2) $r(A_{m \\times n}) \\leq \\min(m,n);$\n",
    "\n",
    "(3) $A \\neq 0 \\Rightarrow r(A) \\geq 1$；\n",
    "\n",
    "(4) $r(A \\pm B) \\leq r(A) + r(B);$\n",
    "\n",
    "(5) 初等变换不改变矩阵的秩\n",
    "\n",
    "(6) $r(A) + r(B) - n \\leq r(AB) \\leq \\min(r(A),r(B)),$特别若$AB = O$\n",
    "则：$r(A) + r(B) \\leq n$\n",
    "\n",
    "(7) 若$A^{- 1}$存在$\\Rightarrow r(AB) = r(B);$ 若$B^{- 1}$存在\n",
    "$\\Rightarrow r(AB) = r(A);$\n",
    "\n",
    "若$r(A_{m \\times n}) = n \\Rightarrow r(AB) = r(B);$ 若$r(A_{m \\times s}) = n\\Rightarrow r(AB) = r\\left( A \\right)$。\n",
    "\n",
    "(8) $r(A_{m \\times s}) = n \\Leftrightarrow Ax = 0$只有零解\n",
    "\n",
    "**8.分块求逆公式**\n",
    "\n",
    "$\\begin{pmatrix} A & O \\\\ O & B \\\\ \\end{pmatrix}^{- 1} = \\begin{pmatrix} A^{-1} & O \\\\ O & B^{- 1} \\\\ \\end{pmatrix}$； $\\begin{pmatrix} A & C \\\\ O & B \\\\\\end{pmatrix}^{- 1} = \\begin{pmatrix} A^{- 1}& - A^{- 1}CB^{- 1} \\\\ O & B^{- 1} \\\\ \\end{pmatrix}$；\n",
    "\n",
    "$\\begin{pmatrix} A & O \\\\ C & B \\\\ \\end{pmatrix}^{- 1} = \\begin{pmatrix}  A^{- 1}&{O} \\\\   - B^{- 1}CA^{- 1} & B^{- 1} \\\\\\end{pmatrix}$； $\\begin{pmatrix} O & A \\\\ B & O \\\\ \\end{pmatrix}^{- 1} =\\begin{pmatrix} O & B^{- 1} \\\\ A^{- 1} & O \\\\ \\end{pmatrix}$\n",
    "\n",
    "这里$A$，$B$均为可逆方阵。\n",
    "\n",
    "#### 向量\n",
    "\n",
    "**1.有关向量组的线性表示**\n",
    "\n",
    "(1)$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}$线性相关$\\Leftrightarrow$至少有一个向量可以用其余向量线性表示。\n",
    "\n",
    "(2)$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}$线性无关，$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}$，$\\beta$线性相关$\\Leftrightarrow \\beta$可以由$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}$唯一线性表示。\n",
    "\n",
    "(3) $\\beta$可以由$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}$线性表示\n",
    "$\\Leftrightarrow r(\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}) =r(\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s},\\beta)$ 。\n",
    "\n",
    "**2.有关向量组的线性相关性**\n",
    "\n",
    "(1)部分相关，整体相关；整体无关，部分无关.\n",
    "\n",
    "(2) ① $n$个$n$维向量\n",
    "$\\alpha_{1},\\alpha_{2}\\cdots\\alpha_{n}$线性无关$\\Leftrightarrow \\left|\\left\\lbrack \\alpha_{1}\\alpha_{2}\\cdots\\alpha_{n} \\right\\rbrack \\right| \\neq0$， $n$个$n$维向量$\\alpha_{1},\\alpha_{2}\\cdots\\alpha_{n}$线性相关\n",
    "$\\Leftrightarrow |\\lbrack\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{n}\\rbrack| = 0$\n",
    "。\n",
    "\n",
    "② $n + 1$个$n$维向量线性相关。\n",
    "\n",
    "③ 若$\\alpha_{1},\\alpha_{2}\\cdots\\alpha_{S}$线性无关，则添加分量后仍线性无关；或一组向量线性相关，去掉某些分量后仍线性相关。\n",
    "\n",
    "**3.有关向量组的线性表示**\n",
    "\n",
    "(1) $\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}$线性相关$\\Leftrightarrow$至少有一个向量可以用其余向量线性表示。\n",
    "\n",
    "(2) $\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}$线性无关，$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}$，$\\beta$线性相关$\\Leftrightarrow\\beta$ 可以由$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}$唯一线性表示。\n",
    "\n",
    "(3) $\\beta$可以由$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}$线性表示\n",
    "$\\Leftrightarrow r(\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}) =r(\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s},\\beta)$\n",
    "\n",
    "**4.向量组的秩与矩阵的秩之间的关系**\n",
    "\n",
    "设$r(A_{m \\times n}) =r$，则$A$的秩$r(A)$与$A$的行列向量组的线性相关性关系为：\n",
    "\n",
    "(1) 若$r(A_{m \\times n}) = r = m$，则$A$的行向量组线性无关。\n",
    "\n",
    "(2) 若$r(A_{m \\times n}) = r < m$，则$A$的行向量组线性相关。\n",
    "\n",
    "(3) 若$r(A_{m \\times n}) = r = n$，则$A$的列向量组线性无关。\n",
    "\n",
    "(4) 若$r(A_{m \\times n}) = r < n$，则$A$的列向量组线性相关。\n",
    "\n",
    "**5.**$\\mathbf{n}$**维向量空间的基变换公式及过渡矩阵**\n",
    "\n",
    "若$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{n}$与$\\beta_{1},\\beta_{2},\\cdots,\\beta_{n}$是向量空间$V$的两组基，则基变换公式为：\n",
    "\n",
    "$(\\beta_{1},\\beta_{2},\\cdots,\\beta_{n}) = (\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{n})\\begin{bmatrix}  c_{11}& c_{12}& \\cdots & c_{1n} \\\\  c_{21}& c_{22}&\\cdots & c_{2n} \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\  c_{n1}& c_{n2} & \\cdots & c_{{nn}} \\\\\\end{bmatrix} = (\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{n})C$\n",
    "\n",
    "其中$C$是可逆矩阵，称为由基$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{n}$到基$\\beta_{1},\\beta_{2},\\cdots,\\beta_{n}$的过渡矩阵。\n",
    "\n",
    "**6.坐标变换公式**\n",
    "\n",
    "若向量$\\gamma$在基$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{n}$与基$\\beta_{1},\\beta_{2},\\cdots,\\beta_{n}$的坐标分别是\n",
    "$X = {(x_{1},x_{2},\\cdots,x_{n})}^{T}$，\n",
    "\n",
    "$Y = \\left( y_{1},y_{2},\\cdots,y_{n} \\right)^{T}$ 即： $\\gamma =x_{1}\\alpha_{1} + x_{2}\\alpha_{2} + \\cdots + x_{n}\\alpha_{n} = y_{1}\\beta_{1} +y_{2}\\beta_{2} + \\cdots + y_{n}\\beta_{n}$，则向量坐标变换公式为$X = CY$ 或$Y = C^{- 1}X$，其中$C$是从基$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{n}$到基$\\beta_{1},\\beta_{2},\\cdots,\\beta_{n}$的过渡矩阵。\n",
    "\n",
    "**7.向量的内积**\n",
    "\n",
    "$(\\alpha,\\beta) = a_{1}b_{1} + a_{2}b_{2} + \\cdots + a_{n}b_{n} = \\alpha^{T}\\beta = \\beta^{T}\\alpha$\n",
    "\n",
    "**8.Schmidt正交化**\n",
    "\n",
    "若$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}$线性无关，则可构造$\\beta_{1},\\beta_{2},\\cdots,\\beta_{s}$使其两两正交，且$\\beta_{i}$仅是$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{i}$的线性组合$(i= 1,2,\\cdots,n)$，再把$\\beta_{i}$单位化，记$\\gamma_{i} =\\frac{\\beta_{i}}{\\left| \\beta_{i}\\right|}$，则$\\gamma_{1},\\gamma_{2},\\cdots,\\gamma_{i}$是规范正交向量组。其中\n",
    "$\\beta_{1} = \\alpha_{1}$， $\\beta_{2} = \\alpha_{2} -\\frac{(\\alpha_{2},\\beta_{1})}{(\\beta_{1},\\beta_{1})}\\beta_{1}$ ， $\\beta_{3} =\\alpha_{3} - \\frac{(\\alpha_{3},\\beta_{1})}{(\\beta_{1},\\beta_{1})}\\beta_{1} -\\frac{(\\alpha_{3},\\beta_{2})}{(\\beta_{2},\\beta_{2})}\\beta_{2}$ ，\n",
    "\n",
    "............\n",
    "\n",
    "$\\beta_{s} = \\alpha_{s} - \\frac{(\\alpha_{s},\\beta_{1})}{(\\beta_{1},\\beta_{1})}\\beta_{1} - \\frac{(\\alpha_{s},\\beta_{2})}{(\\beta_{2},\\beta_{2})}\\beta_{2} - \\cdots - \\frac{(\\alpha_{s},\\beta_{s - 1})}{(\\beta_{s - 1},\\beta_{s - 1})}\\beta_{s - 1}$\n",
    "\n",
    "**9.正交基及规范正交基**\n",
    "\n",
    "向量空间一组基中的向量如果两两正交，就称为正交基；若正交基中每个向量都是单位向量，就称其为规范正交基。\n",
    "\n",
    "#### 线性方程组\n",
    "\n",
    "**1．克莱姆法则**\n",
    "\n",
    "线性方程组$\\begin{cases}  a_{11}x_{1} + a_{12}x_{2} + \\cdots +a_{1n}x_{n} = b_{1} \\\\   a_{21}x_{1} + a_{22}x_{2} + \\cdots + a_{2n}x_{n} =b_{2} \\\\   \\quad\\cdots\\cdots\\cdots\\cdots\\cdots\\cdots\\cdots\\cdots\\cdots \\\\ a_{n1}x_{1} + a_{n2}x_{2} + \\cdots + a_{{nn}}x_{n} = b_{n} \\\\ \\end{cases}$，如果系数行列式$D = \\left| A \\right| \\neq 0$，则方程组有唯一解，$x_{1} = \\frac{D_{1}}{D},x_{2} = \\frac{D_{2}}{D},\\cdots,x_{n} =\\frac{D_{n}}{D}$，其中$D_{j}$是把$D$中第$j$列元素换成方程组右端的常数列所得的行列式。\n",
    "\n",
    "**2.** $n$阶矩阵$A$可逆$\\Leftrightarrow Ax = 0$只有零解。$\\Leftrightarrow\\forall b,Ax = b$总有唯一解，一般地，$r(A_{m \\times n}) = n \\Leftrightarrow Ax= 0$只有零解。\n",
    "\n",
    "**3.非奇次线性方程组有解的充分必要条件，线性方程组解的性质和解的结构**\n",
    "\n",
    "(1) 设$A$为$m \\times n$矩阵，若$r(A_{m \\times n}) = m$，则对$Ax =b$而言必有$r(A) = r(A \\vdots b) = m$，从而$Ax = b$有解。\n",
    "\n",
    "(2) 设$x_{1},x_{2},\\cdots x_{s}$为$Ax = b$的解，则$k_{1}x_{1} + k_{2}x_{2}\\cdots + k_{s}x_{s}$当$k_{1} + k_{2} + \\cdots + k_{s} = 1$时仍为$Ax =b$的解；但当$k_{1} + k_{2} + \\cdots + k_{s} = 0$时，则为$Ax =0$的解。特别$\\frac{x_{1} + x_{2}}{2}$为$Ax = b$的解；$2x_{3} - (x_{1} +x_{2})$为$Ax = 0$的解。\n",
    "\n",
    "(3) 非齐次线性方程组${Ax} = b$无解$\\Leftrightarrow r(A) + 1 =r(\\overline{A}) \\Leftrightarrow b$不能由$A$的列向量$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{n}$线性表示。\n",
    "\n",
    "**4.奇次线性方程组的基础解系和通解，解空间，非奇次线性方程组的通解**\n",
    "\n",
    "(1) 齐次方程组${Ax} = 0$恒有解(必有零解)。当有非零解时，由于解向量的任意线性组合仍是该齐次方程组的解向量，因此${Ax}= 0$的全体解向量构成一个向量空间，称为该方程组的解空间，解空间的维数是$n - r(A)$，解空间的一组基称为齐次方程组的基础解系。\n",
    "\n",
    "(2) $\\eta_{1},\\eta_{2},\\cdots,\\eta_{t}$是${Ax} = 0$的基础解系，即：\n",
    "\n",
    "1) $\\eta_{1},\\eta_{2},\\cdots,\\eta_{t}$是${Ax} = 0$的解；\n",
    "\n",
    "2) $\\eta_{1},\\eta_{2},\\cdots,\\eta_{t}$线性无关；\n",
    "\n",
    "3) ${Ax} = 0$的任一解都可以由$\\eta_{1},\\eta_{2},\\cdots,\\eta_{t}$线性表出.\n",
    "$k_{1}\\eta_{1} + k_{2}\\eta_{2} + \\cdots + k_{t}\\eta_{t}$是${Ax} = 0$的通解，其中$k_{1},k_{2},\\cdots,k_{t}$是任意常数。\n",
    "\n",
    "#### 矩阵的特征值和特征向量\n",
    "\n",
    "**1.矩阵的特征值和特征向量的概念及性质**\n",
    "\n",
    "(1) 设$\\lambda$是$A$的一个特征值，则 ${kA},{aA} + {bE},A^{2},A^{m},f(A),A^{T},A^{- 1},A^{*}$有一个特征值分别为\n",
    "${kλ},{aλ} + b,\\lambda^{2},\\lambda^{m},f(\\lambda),\\lambda,\\lambda^{- 1},\\frac{|A|}{\\lambda},$且对应特征向量相同（$A^{T}$ 例外）。\n",
    "\n",
    "(2)若$\\lambda_{1},\\lambda_{2},\\cdots,\\lambda_{n}$为$A$的$n$个特征值，则$\\sum_{i= 1}^{n}\\lambda_{i} = \\sum_{i = 1}^{n}a_{{ii}},\\prod_{i = 1}^{n}\\lambda_{i}= |A|$ ,从而$|A| \\neq 0 \\Leftrightarrow A$没有特征值。\n",
    "\n",
    "(3)设$\\lambda_{1},\\lambda_{2},\\cdots,\\lambda_{s}$为$A$的$s$个特征值，对应特征向量为$\\alpha_{1},\\alpha_{2},\\cdots,\\alpha_{s}$，\n",
    "\n",
    "若: $\\alpha = k_{1}\\alpha_{1} + k_{2}\\alpha_{2} + \\cdots + k_{s}\\alpha_{s}$ ,\n",
    "\n",
    "则: $A^{n}\\alpha = k_{1}A^{n}\\alpha_{1} + k_{2}A^{n}\\alpha_{2} + \\cdots +k_{s}A^{n}\\alpha_{s} = k_{1}\\lambda_{1}^{n}\\alpha_{1} +k_{2}\\lambda_{2}^{n}\\alpha_{2} + \\cdots k_{s}\\lambda_{s}^{n}\\alpha_{s}$ 。\n",
    "\n",
    "**2.相似变换、相似矩阵的概念及性质**\n",
    "\n",
    "(1) 若$A \\sim B$，则\n",
    "\n",
    "1) $A^{T} \\sim B^{T},A^{- 1} \\sim B^{- 1},,A^{*} \\sim B^{*}$\n",
    "\n",
    "2) $|A| = |B|,\\sum_{i = 1}^{n}A_{{ii}} = \\sum_{i =1}^{n}b_{{ii}},r(A) = r(B)$\n",
    "\n",
    "3) $|\\lambda E - A| = |\\lambda E - B|$，对$\\forall\\lambda$成立\n",
    "\n",
    "**3.矩阵可相似对角化的充分必要条件**\n",
    "\n",
    "(1)设$A$为$n$阶方阵，则$A$可对角化$\\Leftrightarrow$对每个$k_{i}$重根特征值$\\lambda_{i}$，有$n-r(\\lambda_{i}E - A) = k_{i}$\n",
    "\n",
    "(2) 设$A$可对角化，则由$P^{- 1}{AP} = \\Lambda,$有$A = {PΛ}P^{-1}$，从而$A^{n} = P\\Lambda^{n}P^{- 1}$\n",
    "\n",
    "(3) 重要结论\n",
    "\n",
    "1) 若$A \\sim B,C \\sim D$，则$\\begin{bmatrix}  A & O \\\\ O & C \\\\\\end{bmatrix} \\sim \\begin{bmatrix} B & O \\\\  O & D \\\\\\end{bmatrix}$.\n",
    "\n",
    "2) 若$A \\sim B$，则$f(A) \\sim f(B),\\left| f(A) \\right| \\sim \\left| f(B)\\right|$，其中$f(A)$为关于$n$阶方阵$A$的多项式。\n",
    "\n",
    "3) 若$A$为可对角化矩阵，则其非零特征值的个数(重根重复计算)＝秩($A$)\n",
    "\n",
    "**4.实对称矩阵的特征值、特征向量及相似对角阵**\n",
    "\n",
    "(1)相似矩阵：设$A,B$为两个$n$阶方阵，如果存在一个可逆矩阵$P$，使得$B =P^{- 1}{AP}$成立，则称矩阵$A$与$B$相似，记为$A \\sim B$。\n",
    "\n",
    "(2)相似矩阵的性质：如果$A \\sim B$则有：\n",
    "\n",
    "1) $A^{T} \\sim B^{T}$\n",
    "\n",
    "2) $A^{- 1} \\sim B^{- 1}$ （若$A$，$B$均可逆）\n",
    "\n",
    "3) $A^{k} \\sim B^{k}$ （$k$为正整数）\n",
    "\n",
    "4) $\\left| {λE} - A \\right| = \\left| {λE} - B \\right|$，从而$A,B$\n",
    "有相同的特征值\n",
    "\n",
    "5) $\\left| A \\right| = \\left| B \\right|$，从而$A,B$同时可逆或者不可逆\n",
    "\n",
    "6) 秩$\\left( A \\right) =$秩$\\left( B \\right),\\left| {λE} - A \\right| =\\left| {λE} - B \\right|$，$A,B$不一定相似\n",
    "\n",
    "#### 二次型\n",
    "\n",
    "**1.**$\\mathbf{n}$**个变量**$\\mathbf{x}_{\\mathbf{1}}\\mathbf{,}\\mathbf{x}_{\\mathbf{2}}\\mathbf{,\\cdots,}\\mathbf{x}_{\\mathbf{n}}$**的二次齐次函数**\n",
    "\n",
    "$f(x_{1},x_{2},\\cdots,x_{n}) = \\sum_{i = 1}^{n}{\\sum_{j =1}^{n}{a_{{ij}}x_{i}y_{j}}}$，其中$a_{{ij}} = a_{{ji}}(i,j =1,2,\\cdots,n)$，称为$n$元二次型，简称二次型. 若令$x = \\ \\begin{bmatrix}x_{1} \\\\ x_{1} \\\\  \\vdots \\\\ x_{n} \\\\ \\end{bmatrix},A = \\begin{bmatrix}  a_{11}& a_{12}& \\cdots & a_{1n} \\\\  a_{21}& a_{22}& \\cdots & a_{2n} \\\\ \\cdots &\\cdots &\\cdots &\\cdots \\\\  a_{n1}& a_{n2} & \\cdots & a_{{nn}} \\\\\\end{bmatrix}$,这二次型$f$可改写成矩阵向量形式$f =x^{T}{Ax}$。其中$A$称为二次型矩阵，因为$a_{{ij}} =a_{{ji}}(i,j =1,2,\\cdots,n)$，所以二次型矩阵均为对称矩阵，且二次型与对称矩阵一一对应，并把矩阵$A$的秩称为二次型的秩。\n",
    "\n",
    "**2.惯性定理，二次型的标准形和规范形**\n",
    "\n",
    "(1) 惯性定理\n",
    "\n",
    "对于任一二次型，不论选取怎样的合同变换使它化为仅含平方项的标准型，其正负惯性指数与所选变换无关，这就是所谓的惯性定理。\n",
    "\n",
    "(2) 标准形\n",
    "\n",
    "二次型$f = \\left( x_{1},x_{2},\\cdots,x_{n} \\right) =x^{T}{Ax}$经过合同变换$x = {Cy}$化为$f = x^{T}{Ax} =y^{T}C^{T}{AC}$\n",
    "\n",
    "$y = \\sum_{i = 1}^{r}{d_{i}y_{i}^{2}}$称为 $f(r \\leq n)$的标准形。在一般的数域内，二次型的标准形不是唯一的，与所作的合同变换有关，但系数不为零的平方项的个数由$r(A)$唯一确定。\n",
    "\n",
    "(3) 规范形\n",
    "\n",
    "任一实二次型$f$都可经过合同变换化为规范形$f = z_{1}^{2} + z_{2}^{2} + \\cdots z_{p}^{2} - z_{p + 1}^{2} - \\cdots -z_{r}^{2}$，其中$r$为$A$的秩，$p$为正惯性指数，$r -p$为负惯性指数，且规范型唯一。\n",
    "\n",
    "**3.用正交变换和配方法化二次型为标准形，二次型及其矩阵的正定性**\n",
    "\n",
    "设$A$正定$\\Rightarrow {kA}(k > 0),A^{T},A^{- 1},A^{*}$正定；$|A| >0$,$A$可逆；$a_{{ii}} > 0$，且$|A_{{ii}}| > 0$\n",
    "\n",
    "$A$，$B$正定$\\Rightarrow A +B$正定，但${AB}$，${BA}$不一定正定\n",
    "\n",
    "$A$正定$\\Leftrightarrow f(x) = x^{T}{Ax} > 0,\\forall x \\neq 0$\n",
    "\n",
    "$\\Leftrightarrow A$的各阶顺序主子式全大于零\n",
    "\n",
    "$\\Leftrightarrow A$的所有特征值大于零\n",
    "\n",
    "$\\Leftrightarrow A$的正惯性指数为$n$\n",
    "\n",
    "$\\Leftrightarrow$存在可逆阵$P$使$A = P^{T}P$\n",
    "\n",
    "$\\Leftrightarrow$存在正交矩阵$Q$，使$Q^{T}{AQ} = Q^{- 1}{AQ} =\\begin{pmatrix} \\lambda_{1} & & \\\\ \\begin{matrix}  & \\\\  & \\\\ \\end{matrix} &\\ddots & \\\\  & & \\lambda_{n} \\\\ \\end{pmatrix},$\n",
    "\n",
    "其中$\\lambda_{i} > 0,i = 1,2,\\cdots,n.$正定$\\Rightarrow {kA}(k >0),A^{T},A^{- 1},A^{*}$正定； $|A| > 0,A$可逆；$a_{{ii}} >0$，且$|A_{{ii}}| > 0$ 。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b187e113",
   "metadata": {},
   "source": [
    "### 概率论和数理统计\n",
    "\n",
    "#### 随机事件和概率\n",
    "\n",
    "**1.事件的关系与运算**\n",
    "\n",
    "(1) 子事件：$A \\subset B$，若$A$发生，则$B$发生。\n",
    "\n",
    "(2) 相等事件：$A = B$，即$A \\subset B$，且$B \\subset A$ 。\n",
    "\n",
    "(3) 和事件：$A\\bigcup B$（或$A + B$），$A$与$B$中至少有一个发生。\n",
    "\n",
    "(4) 差事件：$A - B$，$A$发生但$B$不发生。\n",
    "\n",
    "(5) 积事件：$A\\bigcap B$（或${AB}$），$A$与$B$同时发生。\n",
    "\n",
    "(6) 互斥事件（互不相容）：$A\\bigcap B$=$\\varnothing$。\n",
    "\n",
    "(7) 互逆事件（对立事件）：\n",
    "$A\\bigcap B=\\varnothing ,A\\bigcup B=\\Omega ,A=\\bar{B},B=\\bar{A}$\n",
    "\n",
    "**2.运算律**\n",
    "\n",
    "(1) 交换律：$A\\bigcup B=B\\bigcup A,A\\bigcap B=B\\bigcap A$\n",
    "\n",
    "(2) 结合律：$(A\\bigcup B)\\bigcup C=A\\bigcup (B\\bigcup C)$\n",
    "\n",
    "(3) 分配律：$(A\\bigcap B)\\bigcap C=A\\bigcap (B\\bigcap C)$\n",
    "\n",
    "**3.德$\\centerdot $摩根律**\n",
    "\n",
    "$\\overline{A\\bigcup B}=\\bar{A}\\bigcap \\bar{B}$                 $\\overline{A\\bigcap B}=\\bar{A}\\bigcup \\bar{B}$\n",
    "\n",
    "**4.完全事件组** \n",
    "\n",
    "${{A}_{1}}{{A}_{2}}\\cdots {{A}_{n}}$两两互斥，且和事件为必然事件，即${{A}_{i}}\\bigcap {{A}_{j}}=\\varnothing, i\\ne j ,\\underset{i=1}{\\overset{n}{\\mathop \\bigcup }}\\,=\\Omega $\n",
    "\n",
    "**5.概率的基本公式**\n",
    "(1)条件概率:\n",
    "\n",
    " $P(B|A)=\\frac{P(AB)}{P(A)}$,表示$A$发生的条件下，$B$发生的概率。\n",
    " \n",
    "(2)全概率公式：\n",
    "\n",
    "$P(A)=\\sum\\limits_{i=1}^{n}{P(A|{{B}_{i}})P({{B}_{i}}),{{B}_{i}}{{B}_{j}}}=\\varnothing ,i\\ne j,\\underset{i=1}{\\overset{n}{\\mathop{\\bigcup }}}\\,{{B}_{i}}=\\Omega $\n",
    "(3) Bayes公式：\n",
    "\n",
    "$P({{B}_{j}}|A)=\\frac{P(A|{{B}_{j}})P({{B}_{j}})}{\\sum\\limits_{i=1}^{n}{P(A|{{B}_{i}})P({{B}_{i}})}},j=1,2,\\cdots ,n$\n",
    "注：上述公式中事件${{B}_{i}}$的个数可为可列个。\n",
    "\n",
    "(4)乘法公式：\n",
    "\n",
    "$P({{A}_{1}}{{A}_{2}})=P({{A}_{1}})P({{A}_{2}}|{{A}_{1}})=P({{A}_{2}})P({{A}_{1}}|{{A}_{2}})$\n",
    "$P({{A}_{1}}{{A}_{2}}\\cdots {{A}_{n}})=P({{A}_{1}})P({{A}_{2}}|{{A}_{1}})P({{A}_{3}}|{{A}_{1}}{{A}_{2}})\\cdots P({{A}_{n}}|{{A}_{1}}{{A}_{2}}\\cdots {{A}_{n-1}})$\n",
    "\n",
    "**6.事件的独立性**\n",
    "\n",
    "(1)$A$与$B$相互独立$\\Leftrightarrow P(AB)=P(A)P(B)$\n",
    "\n",
    "(2)$A$，$B$，$C$两两独立\n",
    "$\\Leftrightarrow P(AB)=P(A)P(B)$;$P(BC)=P(B)P(C)$ ;$P(AC)=P(A)P(C)$;\n",
    "\n",
    "(3)$A$，$B$，$C$相互独立\n",
    "$\\Leftrightarrow P(AB)=P(A)P(B)$;     $P(BC)=P(B)P(C)$ ;\n",
    "$P(AC)=P(A)P(C)$  ;   $P(ABC)=P(A)P(B)P(C)$\n",
    "\n",
    "**7.独立重复试验** \n",
    "\n",
    "将某试验独立重复$n$次，若每次实验中事件A发生的概率为$p$，则$n$次试验中$A$发生$k$次的概率为：\n",
    "$P(X=k)=C_{n}^{k}{{p}^{k}}{{(1-p)}^{n-k}}$\n",
    "\n",
    "**8.重要公式与结论**\n",
    "\n",
    "$(1)P(\\bar{A})=1-P(A)$\n",
    "\n",
    "$(2)P(A\\bigcup B)=P(A)+P(B)-P(AB)$\n",
    "   $P(A\\bigcup B\\bigcup C)=P(A)+P(B)+P(C)-P(AB)-P(BC)-P(AC)+P(ABC)$\n",
    "   \n",
    "$(3)P(A-B)=P(A)-P(AB)$\n",
    "\n",
    "$(4)P(A\\bar{B})=P(A)-P(AB),P(A)=P(AB)+P(A\\bar{B}),$\n",
    "\n",
    " $P(A\\bigcup B)=P(A)+P(\\bar{A}B)=P(AB)+P(A\\bar{B})+P(\\bar{A}B)$\n",
    " \n",
    "(5)条件概率$P(\\centerdot |B)$满足概率的所有性质，\n",
    "例如：. $P({{\\bar{A}}_{1}}|B)=1-P({{A}_{1}}|B)$\n",
    "$P({{A}_{1}}\\bigcup {{A}_{2}}|B)=P({{A}_{1}}|B)+P({{A}_{2}}|B)-P({{A}_{1}}{{A}_{2}}|B)$\n",
    "$P({{A}_{1}}{{A}_{2}}|B)=P({{A}_{1}}|B)P({{A}_{2}}|{{A}_{1}}B)$\n",
    "\n",
    "(6)若${{A}_{1}},{{A}_{2}},\\cdots ,{{A}_{n}}$相互独立，则$P(\\bigcap\\limits_{i=1}^{n}{{{A}_{i}}})=\\prod\\limits_{i=1}^{n}{P({{A}_{i}})},$\n",
    " $P(\\bigcup\\limits_{i=1}^{n}{{{A}_{i}}})=\\prod\\limits_{i=1}^{n}{(1-P({{A}_{i}}))}$\n",
    "\n",
    "(7)互斥、互逆与独立性之间的关系：\n",
    "$A$与$B$互逆$\\Rightarrow$ $A$与$B$互斥，但反之不成立，$A$与$B$互斥（或互逆）且均非零概率事件$\\Rightarrow $$A$与$B$不独立.\n",
    "\n",
    "(8)若${{A}_{1}},{{A}_{2}},\\cdots ,{{A}_{m}},{{B}_{1}},{{B}_{2}},\\cdots ,{{B}_{n}}$相互独立，则$f({{A}_{1}},{{A}_{2}},\\cdots ,{{A}_{m}})$与$g({{B}_{1}},{{B}_{2}},\\cdots ,{{B}_{n}})$也相互独立，其中$f(\\centerdot ),g(\\centerdot )$分别表示对相应事件做任意事件运算后所得的事件，另外，概率为1（或0）的事件与任何事件相互独立.\n",
    "\n",
    "\n",
    "\n",
    "#### 随机变量及其概率分布\n",
    "\n",
    "**1.随机变量及概率分布**\n",
    "\n",
    "取值带有随机性的变量，严格地说是定义在样本空间上，取值于实数的函数称为随机变量，概率分布通常指分布函数或分布律\n",
    "\n",
    "**2.分布函数的概念与性质**\n",
    "\n",
    "定义： $F(x) = P(X \\leq x), - \\infty < x < + \\infty$\n",
    "\n",
    "性质：(1)$0 \\leq F(x) \\leq 1$ \n",
    "\n",
    "(2) $F(x)$单调不减\n",
    "\n",
    "(3) 右连续$F(x + 0) = F(x)$ \n",
    "\n",
    "(4) $F( - \\infty) = 0,F( + \\infty) = 1$\n",
    "\n",
    "**3.离散型随机变量的概率分布**\n",
    "\n",
    "$P(X = x_{i}) = p_{i},i = 1,2,\\cdots,n,\\cdots\\quad\\quad p_{i} \\geq 0,\\sum_{i =1}^{\\infty}p_{i} = 1$\n",
    "\n",
    "**4.连续型随机变量的概率密度**\n",
    "\n",
    "概率密度$f(x)$;非负可积，且:\n",
    "\n",
    "(1)$f(x) \\geq 0,$ \n",
    "\n",
    "(2)$\\int_{- \\infty}^{+\\infty}{f(x){dx} = 1}$ \n",
    "\n",
    "(3)$x$为$f(x)$的连续点，则:\n",
    "\n",
    "$f(x) = F'(x)$分布函数$F(x) = \\int_{- \\infty}^{x}{f(t){dt}}$\n",
    "\n",
    "**5.常见分布**\n",
    "\n",
    "(1) 0-1分布:$P(X = k) = p^{k}{(1 - p)}^{1 - k},k = 0,1$\n",
    "\n",
    "(2) 二项分布:$B(n,p)$： $P(X = k) = C_{n}^{k}p^{k}{(1 - p)}^{n - k},k =0,1,\\cdots,n$\n",
    "\n",
    "(3) **Poisson**分布:$p(\\lambda)$： $P(X = k) = \\frac{\\lambda^{k}}{k!}e^{-\\lambda},\\lambda > 0,k = 0,1,2\\cdots$\n",
    "\n",
    "(4) 均匀分布$U(a,b)$：$f(x) = \\{ \\begin{matrix}  & \\frac{1}{b - a},a < x< b \\\\  & 0, \\\\ \\end{matrix} $\n",
    "\n",
    "(5) 正态分布:$N(\\mu,\\sigma^{2}):$ $\\varphi(x) =\\frac{1}{\\sqrt{2\\pi}\\sigma}e^{- \\frac{{(x - \\mu)}^{2}}{2\\sigma^{2}}},\\sigma > 0,\\infty < x < + \\infty$\n",
    "\n",
    "(6)指数分布:$E(\\lambda):f(x) =\\{ \\begin{matrix}  & \\lambda e^{-{λx}},x > 0,\\lambda > 0 \\\\  & 0, \\\\ \\end{matrix} $\n",
    "\n",
    "(7)几何分布:$G(p):P(X = k) = {(1 - p)}^{k - 1}p,0 < p < 1,k = 1,2,\\cdots.$\n",
    "\n",
    "(8)超几何分布: $H(N,M,n):P(X = k) = \\frac{C_{M}^{k}C_{N - M}^{n -k}}{C_{N}^{n}},k =0,1,\\cdots,min(n,M)$\n",
    "\n",
    "**6.随机变量函数的概率分布**\n",
    "\n",
    "(1)离散型：$P(X = x_{1}) = p_{i},Y = g(X)$\n",
    "\n",
    "则: $P(Y = y_{j}) = \\sum_{g(x_{i}) = y_{i}}^{}{P(X = x_{i})}$\n",
    "\n",
    "(2)连续型：$X\\tilde{\\ }f_{X}(x),Y = g(x)$\n",
    "\n",
    "则:$F_{y}(y) = P(Y \\leq y) = P(g(X) \\leq y) = \\int_{g(x) \\leq y}^{}{f_{x}(x)dx}$， $f_{Y}(y) = F'_{Y}(y)$\n",
    "\n",
    "**7.重要公式与结论**\n",
    "\n",
    "(1) $X\\sim N(0,1) \\Rightarrow \\varphi(0) = \\frac{1}{\\sqrt{2\\pi}},\\Phi(0) =\\frac{1}{2},$ $\\Phi( - a) = P(X \\leq - a) = 1 - \\Phi(a)$\n",
    "\n",
    "(2) $X\\sim N\\left( \\mu,\\sigma^{2} \\right) \\Rightarrow \\frac{X -\\mu}{\\sigma}\\sim N\\left( 0,1 \\right),P(X \\leq a) = \\Phi(\\frac{a -\\mu}{\\sigma})$\n",
    "\n",
    "(3) $X\\sim E(\\lambda) \\Rightarrow P(X > s + t|X > s) = P(X > t)$\n",
    "\n",
    "(4) $X\\sim G(p) \\Rightarrow P(X = m + k|X > m) = P(X = k)$\n",
    "\n",
    "(5) 离散型随机变量的分布函数为阶梯间断函数；连续型随机变量的分布函数为连续函数，但不一定为处处可导函数。\n",
    "\n",
    "(6) 存在既非离散也非连续型随机变量。\n",
    "\n",
    "#### 多维随机变量及其分布\n",
    "\n",
    "**1.二维随机变量及其联合分布**\n",
    "\n",
    "由两个随机变量构成的随机向量$(X,Y)$， 联合分布为$F(x,y) = P(X \\leq x,Y \\leq y)$\n",
    "\n",
    "**2.二维离散型随机变量的分布**\n",
    "\n",
    "(1) 联合概率分布律 $P\\{ X = x_{i},Y = y_{j}\\} = p_{{ij}};i,j =1,2,\\cdots$\n",
    "\n",
    "(2) 边缘分布律 $p_{i \\cdot} = \\sum_{j = 1}^{\\infty}p_{{ij}},i =1,2,\\cdots$ $p_{\\cdot j} = \\sum_{i}^{\\infty}p_{{ij}},j = 1,2,\\cdots$\n",
    "\n",
    "(3) 条件分布律 $P\\{ X = x_{i}|Y = y_{j}\\} = \\frac{p_{{ij}}}{p_{\\cdot j}}$\n",
    "$P\\{ Y = y_{j}|X = x_{i}\\} = \\frac{p_{{ij}}}{p_{i \\cdot}}$\n",
    "\n",
    "**3. 二维连续性随机变量的密度**\n",
    "\n",
    "(1) 联合概率密度$f(x,y):$\n",
    "\n",
    "1) $f(x,y) \\geq 0$ \n",
    "\n",
    "2) $\\int_{- \\infty}^{+ \\infty}{\\int_{- \\infty}^{+ \\infty}{f(x,y)dxdy}} = 1$\n",
    "\n",
    "(2) 分布函数：$F(x,y) = \\int_{- \\infty}^{x}{\\int_{- \\infty}^{y}{f(u,v)dudv}}$\n",
    "\n",
    "(3) 边缘概率密度： $f_{X}\\left( x \\right) = \\int_{- \\infty}^{+ \\infty}{f\\left( x,y \\right){dy}}$ $f_{Y}(y) = \\int_{- \\infty}^{+ \\infty}{f(x,y)dx}$\n",
    "\n",
    "(4) 条件概率密度：$f_{X|Y}\\left( x \\middle| y \\right) = \\frac{f\\left( x,y \\right)}{f_{Y}\\left( y \\right)}$ $f_{Y|X}(y|x) = \\frac{f(x,y)}{f_{X}(x)}$\n",
    "\n",
    "**4.常见二维随机变量的联合分布**\n",
    "\n",
    "(1) 二维均匀分布：$(x,y) \\sim U(D)$ ,$f(x,y) = \\begin{cases} \\frac{1}{S(D)},(x,y) \\in D \\\\   0,其他  \\end{cases}$\n",
    "\n",
    "(2) 二维正态分布：$(X,Y)\\sim N(\\mu_{1},\\mu_{2},\\sigma_{1}^{2},\\sigma_{2}^{2},\\rho)$,$(X,Y)\\sim N(\\mu_{1},\\mu_{2},\\sigma_{1}^{2},\\sigma_{2}^{2},\\rho)$\n",
    "\n",
    "$f(x,y) = \\frac{1}{2\\pi\\sigma_{1}\\sigma_{2}\\sqrt{1 - \\rho^{2}}}.\\exp\\left\\{ \\frac{- 1}{2(1 - \\rho^{2})}\\lbrack\\frac{{(x - \\mu_{1})}^{2}}{\\sigma_{1}^{2}} - 2\\rho\\frac{(x - \\mu_{1})(y - \\mu_{2})}{\\sigma_{1}\\sigma_{2}} + \\frac{{(y - \\mu_{2})}^{2}}{\\sigma_{2}^{2}}\\rbrack \\right\\}$\n",
    "\n",
    "**5.随机变量的独立性和相关性**\n",
    "\n",
    "$X$和$Y$的相互独立:$\\Leftrightarrow F\\left( x,y \\right) = F_{X}\\left( x \\right)F_{Y}\\left( y \\right)$:\n",
    "\n",
    "$\\Leftrightarrow p_{{ij}} = p_{i \\cdot} \\cdot p_{\\cdot j}$（离散型）\n",
    "$\\Leftrightarrow f\\left( x,y \\right) = f_{X}\\left( x \\right)f_{Y}\\left( y \\right)$（连续型）\n",
    "\n",
    "$X$和$Y$的相关性：\n",
    "\n",
    "相关系数$\\rho_{{XY}} = 0$时，称$X$和$Y$不相关，\n",
    "否则称$X$和$Y$相关\n",
    "\n",
    "**6.两个随机变量简单函数的概率分布**\n",
    "\n",
    "离散型： $P\\left( X = x_{i},Y = y_{i} \\right) = p_{{ij}},Z = g\\left( X,Y \\right)$ 则：\n",
    "\n",
    "$P(Z = z_{k}) = P\\left\\{ g\\left( X,Y \\right) = z_{k} \\right\\} = \\sum_{g\\left( x_{i},y_{i} \\right) = z_{k}}^{}{P\\left( X = x_{i},Y = y_{j} \\right)}$\n",
    "\n",
    "连续型： $\\left( X,Y \\right) \\sim f\\left( x,y \\right),Z = g\\left( X,Y \\right)$\n",
    "则：\n",
    "\n",
    "$F_{z}\\left( z \\right) = P\\left\\{ g\\left( X,Y \\right) \\leq z \\right\\} = \\iint_{g(x,y) \\leq z}^{}{f(x,y)dxdy}$，$f_{z}(z) = F'_{z}(z)$\n",
    "\n",
    "**7.重要公式与结论**\n",
    "\n",
    "(1) 边缘密度公式： $f_{X}(x) = \\int_{- \\infty}^{+ \\infty}{f(x,y)dy,}$\n",
    "$f_{Y}(y) = \\int_{- \\infty}^{+ \\infty}{f(x,y)dx}$\n",
    "\n",
    "(2) $P\\left\\{ \\left( X,Y \\right) \\in D \\right\\} = \\iint_{D}^{}{f\\left( x,y \\right){dxdy}}$\n",
    "\n",
    "(3) 若$(X,Y)$服从二维正态分布$N(\\mu_{1},\\mu_{2},\\sigma_{1}^{2},\\sigma_{2}^{2},\\rho)$\n",
    "则有：\n",
    "\n",
    "1) $X\\sim N\\left( \\mu_{1},\\sigma_{1}^{2} \\right),Y\\sim N(\\mu_{2},\\sigma_{2}^{2}).$\n",
    "\n",
    "2) $X$与$Y$相互独立$\\Leftrightarrow \\rho = 0$，即$X$与$Y$不相关。\n",
    "\n",
    "3) $C_{1}X + C_{2}Y\\sim N(C_{1}\\mu_{1} + C_{2}\\mu_{2},C_{1}^{2}\\sigma_{1}^{2} + C_{2}^{2}\\sigma_{2}^{2} + 2C_{1}C_{2}\\sigma_{1}\\sigma_{2}\\rho)$\n",
    "\n",
    "4) ${\\ X}$关于$Y=y$的条件分布为： $N(\\mu_{1} + \\rho\\frac{\\sigma_{1}}{\\sigma_{2}}(y - \\mu_{2}),\\sigma_{1}^{2}(1 - \\rho^{2}))$\n",
    "\n",
    "5) $Y$关于$X = x$的条件分布为： $N(\\mu_{2} + \\rho\\frac{\\sigma_{2}}{\\sigma_{1}}(x - \\mu_{1}),\\sigma_{2}^{2}(1 - \\rho^{2}))$\n",
    "\n",
    "(4) 若$X$与$Y$独立，且分别服从$N(\\mu_{1},\\sigma_{1}^{2}),N(\\mu_{1},\\sigma_{2}^{2}),$\n",
    "则：$\\left( X,Y \\right)\\sim N(\\mu_{1},\\mu_{2},\\sigma_{1}^{2},\\sigma_{2}^{2},0),$\n",
    "\n",
    "$C_{1}X + C_{2}Y\\tilde{\\ }N(C_{1}\\mu_{1} + C_{2}\\mu_{2},C_{1}^{2}\\sigma_{1}^{2} C_{2}^{2}\\sigma_{2}^{2}).$\n",
    "\n",
    "(5) 若$X$与$Y$相互独立，$f\\left( x \\right)$和$g\\left( x \\right)$为连续函数， 则$f\\left( X \\right)$和$g(Y)$也相互独立。\n",
    "\n",
    "#### 随机变量的数字特征\n",
    "\n",
    "**1.数学期望**\n",
    "\n",
    "离散型：$P\\left\\{ X = x_{i} \\right\\} = p_{i},E(X) = \\sum_{i}^{}{x_{i}p_{i}}$；\n",
    "\n",
    "连续型： $X\\sim f(x),E(X) = \\int_{- \\infty}^{+ \\infty}{xf(x)dx}$\n",
    "\n",
    "性质：\n",
    "\n",
    "(1) $E(C) = C,E\\lbrack E(X)\\rbrack = E(X)$\n",
    "\n",
    "(2) $E(C_{1}X + C_{2}Y) = C_{1}E(X) + C_{2}E(Y)$\n",
    "\n",
    "(3) 若$X$和$Y$独立，则$E(XY) = E(X)E(Y)$ \n",
    "\n",
    "(4)$\\left\\lbrack E(XY) \\right\\rbrack^{2} \\leq E(X^{2})E(Y^{2})$\n",
    "\n",
    "**2.方差**：$D(X) = E\\left\\lbrack X - E(X) \\right\\rbrack^{2} = E(X^{2}) - \\left\\lbrack E(X) \\right\\rbrack^{2}$\n",
    "\n",
    "**3.标准差**：$\\sqrt{D(X)}$，\n",
    "\n",
    "**4.离散型：**$D(X) = \\sum_{i}^{}{\\left\\lbrack x_{i} - E(X) \\right\\rbrack^{2}p_{i}}$\n",
    "\n",
    "**5.连续型：**$D(X) = {\\int_{- \\infty}^{+ \\infty}\\left\\lbrack x - E(X) \\right\\rbrack}^{2}f(x)dx$\n",
    "\n",
    "性质：\n",
    "\n",
    "(1)$\\ D(C) = 0,D\\lbrack E(X)\\rbrack = 0,D\\lbrack D(X)\\rbrack = 0$\n",
    "\n",
    "(2) $X$与$Y$相互独立，则$D(X \\pm Y) = D(X) + D(Y)$\n",
    "\n",
    "(3)$\\ D\\left( C_{1}X + C_{2} \\right) = C_{1}^{2}D\\left( X \\right)$\n",
    "\n",
    "(4) 一般有 $D(X \\pm Y) = D(X) + D(Y) \\pm 2Cov(X,Y) = D(X) + D(Y) \\pm 2\\rho\\sqrt{D(X)}\\sqrt{D(Y)}$\n",
    "\n",
    "(5)$\\ D\\left( X \\right) < E\\left( X - C \\right)^{2},C \\neq E\\left( X \\right)$\n",
    "\n",
    "(6)$\\ D(X) = 0 \\Leftrightarrow P\\left\\{ X = C \\right\\} = 1$\n",
    "\n",
    "**6.随机变量函数的数学期望**\n",
    "\n",
    "(1) 对于函数$Y = g(x)$\n",
    "\n",
    "$X$为离散型：$P\\{ X = x_{i}\\} = p_{i},E(Y) = \\sum_{i}^{}{g(x_{i})p_{i}}$；\n",
    "\n",
    "$X$为连续型：$X\\sim f(x),E(Y) = \\int_{- \\infty}^{+ \\infty}{g(x)f(x)dx}$\n",
    "\n",
    "(2) $Z = g(X,Y)$;$\\left( X,Y \\right)\\sim P\\{ X = x_{i},Y = y_{j}\\} = p_{{ij}}$; $E(Z) = \\sum_{i}^{}{\\sum_{j}^{}{g(x_{i},y_{j})p_{{ij}}}}$ $\\left( X,Y \\right)\\sim f(x,y)$;$E(Z) = \\int_{- \\infty}^{+ \\infty}{\\int_{- \\infty}^{+ \\infty}{g(x,y)f(x,y)dxdy}}$\n",
    "\n",
    "**7.协方差** \n",
    "\n",
    "$Cov(X,Y) = E\\left\\lbrack (X - E(X)(Y - E(Y)) \\right\\rbrack$\n",
    "\n",
    "**8.相关系数**\n",
    "\n",
    " $\\rho_{{XY}} = \\frac{Cov(X,Y)}{\\sqrt{D(X)}\\sqrt{D(Y)}}$,$k$阶原点矩 $E(X^{k})$;\n",
    "$k$阶中心矩 $E\\left\\{ {\\lbrack X - E(X)\\rbrack}^{k} \\right\\}$\n",
    "\n",
    "性质：\n",
    "\n",
    "(1)$\\ Cov(X,Y) = Cov(Y,X)$\n",
    "\n",
    "(2)$\\ Cov(aX,bY) = abCov(Y,X)$\n",
    "\n",
    "(3)$\\ Cov(X_{1} + X_{2},Y) = Cov(X_{1},Y) + Cov(X_{2},Y)$\n",
    "\n",
    "(4)$\\ \\left| \\rho\\left( X,Y \\right) \\right| \\leq 1$\n",
    "\n",
    "(5) $\\ \\rho\\left( X,Y \\right) = 1 \\Leftrightarrow P\\left( Y = aX + b \\right) = 1$ ，其中$a > 0$\n",
    "\n",
    "$\\rho\\left( X,Y \\right) = - 1 \\Leftrightarrow P\\left( Y = aX + b \\right) = 1$\n",
    "，其中$a < 0$\n",
    "\n",
    "**9.重要公式与结论**\n",
    "\n",
    "(1)$\\ D(X) = E(X^{2}) - E^{2}(X)$\n",
    "\n",
    "(2)$\\ Cov(X,Y) = E(XY) - E(X)E(Y)$\n",
    "\n",
    "(3) $\\left| \\rho\\left( X,Y \\right) \\right| \\leq 1,$且 $\\rho\\left( X,Y \\right) = 1 \\Leftrightarrow P\\left( Y = aX + b \\right) = 1$，其中$a > 0$\n",
    "\n",
    "$\\rho\\left( X,Y \\right) = - 1 \\Leftrightarrow P\\left( Y = aX + b \\right) = 1$，其中$a < 0$\n",
    "\n",
    "(4) 下面5个条件互为充要条件：\n",
    "\n",
    "$\\rho(X,Y) = 0$ $\\Leftrightarrow Cov(X,Y) = 0$ $\\Leftrightarrow E(X,Y) = E(X)E(Y)$ $\\Leftrightarrow D(X + Y) = D(X) + D(Y)$ $\\Leftrightarrow  D(X - Y) = D(X) + D(Y)$\n",
    "\n",
    "注：$X$与$Y$独立为上述5个条件中任何一个成立的充分条件，但非必要条件。\n",
    "\n",
    "#### 数理统计的基本概念\n",
    "\n",
    "**1.基本概念**\n",
    "\n",
    "总体：研究对象的全体，它是一个随机变量，用$X$表示。\n",
    "\n",
    "个体：组成总体的每个基本元素。\n",
    "\n",
    "简单随机样本：来自总体$X$的$n$个相互独立且与总体同分布的随机变量$X_{1},X_{2}\\cdots,X_{n}$，称为容量为$n$的简单随机样本，简称样本。\n",
    "\n",
    "统计量：设$X_{1},X_{2}\\cdots,X_{n},$是来自总体$X$的一个样本，$g(X_{1},X_{2}\\cdots,X_{n})$）是样本的连续函数，且$g()$中不含任何未知参数，则称$g(X_{1},X_{2}\\cdots,X_{n})$为统计量。\n",
    "\n",
    "样本均值：$\\overline{X} = \\frac{1}{n}\\sum_{i = 1}^{n}X_{i}$\n",
    "\n",
    "样本方差：$S^{2} = \\frac{1}{n - 1}\\sum_{i = 1}^{n}{(X_{i} - \\overline{X})}^{2}$\n",
    "\n",
    "样本矩：样本$k$阶原点矩：$A_{k} = \\frac{1}{n}\\sum_{i = 1}^{n}X_{i}^{k},k = 1,2,\\cdots$\n",
    "\n",
    "样本$k$阶中心矩：$B_{k} = \\frac{1}{n}\\sum_{i = 1}^{n}{(X_{i} - \\overline{X})}^{k},k = 1,2,\\cdots$\n",
    "\n",
    "**2.分布**\n",
    "\n",
    "$\\chi^{2}$分布：$\\chi^{2} = X_{1}^{2} + X_{2}^{2} + \\cdots + X_{n}^{2}\\sim\\chi^{2}(n)$，其中$X_{1},X_{2}\\cdots,X_{n},$相互独立，且同服从$N(0,1)$\n",
    "\n",
    "$t$分布：$T = \\frac{X}{\\sqrt{Y/n}}\\sim t(n)$ ，其中$X\\sim N\\left( 0,1 \\right),Y\\sim\\chi^{2}(n),$且$X$，$Y$ 相互独立。\n",
    "\n",
    "$F$分布：$F = \\frac{X/n_{1}}{Y/n_{2}}\\sim F(n_{1},n_{2})$，其中$X\\sim\\chi^{2}\\left( n_{1} \\right),Y\\sim\\chi^{2}(n_{2}),$且$X$，$Y$相互独立。\n",
    "\n",
    "分位数：若$P(X \\leq x_{\\alpha}) = \\alpha,$则称$x_{\\alpha}$为$X$的$\\alpha$分位数\n",
    "\n",
    "**3.正态总体的常用样本分布**\n",
    "\n",
    "(1) 设$X_{1},X_{2}\\cdots,X_{n}$为来自正态总体$N(\\mu,\\sigma^{2})$的样本，\n",
    "\n",
    "$\\overline{X} = \\frac{1}{n}\\sum_{i = 1}^{n}X_{i},S^{2} = \\frac{1}{n - 1}\\sum_{i = 1}^{n}{{(X_{i} - \\overline{X})}^{2},}$则：\n",
    "\n",
    "1) $\\overline{X}\\sim N\\left( \\mu,\\frac{\\sigma^{2}}{n} \\right){\\ \\ }$或者$\\frac{\\overline{X} - \\mu}{\\frac{\\sigma}{\\sqrt{n}}}\\sim N(0,1)$\n",
    "\n",
    "2) $\\frac{(n - 1)S^{2}}{\\sigma^{2}} = \\frac{1}{\\sigma^{2}}\\sum_{i = 1}^{n}{{(X_{i} - \\overline{X})}^{2}\\sim\\chi^{2}(n - 1)}$\n",
    "\n",
    "3) $\\frac{1}{\\sigma^{2}}\\sum_{i = 1}^{n}{{(X_{i} - \\mu)}^{2}\\sim\\chi^{2}(n)}$\n",
    "\n",
    "4)${\\ \\ }\\frac{\\overline{X} - \\mu}{S/\\sqrt{n}}\\sim t(n - 1)$\n",
    "\n",
    "**4.重要公式与结论**\n",
    "\n",
    "(1) 对于$\\chi^{2}\\sim\\chi^{2}(n)$，有$E(\\chi^{2}(n)) = n,D(\\chi^{2}(n)) = 2n;$\n",
    "\n",
    "(2) 对于$T\\sim t(n)$，有$E(T) = 0,D(T) = \\frac{n}{n - 2}(n > 2)$；\n",
    "\n",
    "(3) 对于$F\\tilde{\\ }F(m,n)$，有 $\\frac{1}{F}\\sim F(n,m),F_{a/2}(m,n) = \\frac{1}{F_{1 - a/2}(n,m)};$\n",
    "\n",
    "(4) 对于任意总体$X$，有 $E(\\overline{X}) = E(X),E(S^{2}) = D(X),D(\\overline{X}) = \\frac{D(X)}{n}$"
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