只要掌握这三组公式,便可以在AI学习中如鱼得水了
本文摘录自 百度AI 新手学习必备数学知识 ,为了便于查找和学习。主要包括有微积分、线性代数、概率与数理统计。
数学基础知识
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。
下面是常见的一些数学基础概念,建议大家收藏后再仔细阅读,遇到不懂的概念可以直接在这里查~
§01 高等数学
1.1 导数定义
1.1.1 导数和微分的概念
f ′ ( x 0 ) = lim Δ x → 0 f ( x 0 + Δ x ) − f ( x 0 ) Δ x f'({{x}_{0}})=\underset{\Delta x\to 0}{\mathop{\lim }}\,\frac{f({{x}_{0}}+\Delta x)-f({{x}_{0}})}{\Delta x} f′(x0)=Δx→0limΔxf(x0+Δx)−f(x0)
或者:
f ′ ( x 0 ) = lim x → x 0 f ( x ) − f ( x 0 ) x − x 0 f'({{x}_{0}})=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}} f′(x0)=x→x0limx−x0f(x)−f(x0)
1.1.2 左右导数的几何与物理意义
函数 f ( x ) f(x) f(x)在 x 0 x_0 x0处的左、右导数分别定义为:
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左导数: f ′ − ( x 0 ) = lim Δ x → 0 − f ( x 0 + Δ x ) − f ( x 0 ) Δ x = lim x → x 0 − f ( x ) − f ( x 0 ) x − x 0 , ( x = x 0 + Δ x ) {{{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) f′−(x0)=Δx→0−limΔxf(x0+Δx)−f(x0)=x→x0−limx−x0f(x)−f(x0),(x=x0+Δx)
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右导数: f ′ + ( x 0 ) = lim Δ x → 0 + f ( x 0 + Δ x ) − f ( x 0 ) Δ x = lim x → x 0 + f ( x ) − f ( x 0 ) x − x 0 {{{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}}} f′+(x0)=Δx→0+limΔxf(x0+Δx)−f(x0)=x→x0+limx−x0f(x)−f(x0)
1.1.3 可导性与连续性之间的关系
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Th1: 函数 f ( x ) f(x) f(x) 在 x 0 x_0 x0 处可微 ⇔ f ( x ) \Leftrightarrow f(x) ⇔f(x)在 x 0 x_0 x0 处可导
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Th2: 若函数在点 x 0 x_0 x0处可导,则 y = f ( x ) y=f(x) y=f(x) 在点 x 0 x_0 x0 处连续,反之则不成立。即函数连续不一定可导。
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Th3: f ′ ( x 0 ) {f}'({{x}_{0}}) f′(x0)存在 ⇔ f ′ − ( x 0 ) = f ′ + ( x 0 ) \Leftrightarrow {{{f}'}_{-}}({{x}_{0}})={{{f}'}_{+}}({{x}_{0}}) ⇔f′−(x0)=f′+(x0)
1.1.4 平面曲线的切线和法线
- 切线方程 : y − y 0 = f ′ ( x 0 ) ( x − x 0 ) y-{{y}_{0}}=f'({{x}_{0}})(x-{{x}_{0}}) y−y0=f′(x0)(x−x0)
- 法线方程: y − y 0 = − 1 f ′ ( x 0 ) ( x − x 0 ) , f ′ ( x 0 ) ≠ 0 y-{{y}_{0}}=-\frac{1}{f'({{x}_{0}})}(x-{{x}_{0}}),f'({{x}_{0}})\ne 0 y−y0=−f′(x0)1(x−x0),f′(x0)=0
1.2 微分四则运算
1.2.1 四则运算
设函数 u = u ( x ) , v = v ( x ) u=u(x),v=v(x) u=u(x),v=v(x)]在点 x x x可导则
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\left( {u \pm v} \right)^\prime = u' \pm v',\,\,\,\,d\left( {u \pm v} \right) = du \pm dv
(u±v)′=u′±v′,d(u±v)=du±dv
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(uv{)}'=u{v}'+v{u}',\,\,\,\,d(uv)=udv+vdu
(uv)′=uv′+vu′,d(uv)=udv+vdu
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(\frac{u}{v}{)}'=\frac{v{u}'-u{v}'}{{{v}^{2}}}(v\ne 0),\,\,\,\,d(\frac{u}{v})=\frac{vdu-udv}{{{v}^{2}}}
(vu)′=v2vu′−uv′(v=0),d(vu)=v2vdu−udv
1.2.2 基本导数与微分表
y = c , y ′ = 0 , d y = 0 y = c,\,\,\,\,\,\,y' = 0,\,\,\,dy = 0 y=c,y′=0,dy=0
y = x α , y ′ = α x α − 1 , d y = α x α − 1 d x y = x^\alpha ,\,\,\,\,y' = \alpha x^{\alpha - 1} ,\,\,\,\,dy = \alpha x^{\alpha - 1} dx y=xα,y′=αxα−1,dy=αxα−1dx
y = a x , y ′ = a x ln a , d y = a x ln a ⋅ d x y = a^x ,\,\,\,\,y' = a^x \ln a,\,\,\,\,dy = a^x \ln a \cdot dx y=ax,y′=axlna,dy=axlna⋅dx
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特例: ( e x ) ′ = e x ({{{e}}^{x}}{)}'={{{e}}^{x}} (ex)′=ex d ( e x ) = e x d x d({{{e}}^{x}})={{{e}}^{x}}dx d(ex)=exdx
y = log a x , y ′ = 1 x ln a , d y = 1 x ln a d x y = \log _a x,\,\,\,\,y' = {1 \over {x\ln a}},\,\,\,\,\,dy = {1 \over {x\ln a}}dx y=logax,y′=xlna1,dy=xlna1dx -
特例: y = ln x y=\ln x y=lnx ( ln x ) ′ = 1 x (\ln x{)}'=\frac{1}{x} (lnx)′=x1 d ( ln x ) = 1 x d x d(\ln x)=\frac{1}{x}dx d(lnx)=x1dx
y = sin x , y ′ = cos x , d ( sin x ) = cos x ⋅ d x y = \sin x,\,\,\,\,y' = \cos x,\,\,\,\,d\left( {\sin x} \right) = \cos x \cdot dx y=sinx,y′=cosx,d(sinx)=cosx⋅dx
y = cos x , y ′ = − sin x , d ( cos x ) = − sin x ⋅ d x y = \cos x,\,\,\,\,\,y' = - \sin x,\,\,\,\,d\left( {\cos x} \right) = - \sin x \cdot dx y=cosx,y′=−sinx,d(cosx)=−sinx⋅dx
y = tan x , y ′ = 1 cos 2 x = sec 2 x , d ( tan x ) = sec 2 x ⋅ d x y = \tan x,\,\,\,\,y' = {1 \over {\cos ^2 x}} = \sec ^2 x,\,\,\,\,d\left( {\tan x} \right) = \sec ^2 x \cdot dx y=tanx,y′=cos2x1=sec2x,d(tanx)=sec2x⋅dx
y = cot x , y ′ = − 1 sin 2 x = − csc 2 x , d ( cot x ) = − csc 2 x ⋅ d x y = \cot x,\,\,\,\,y' = - {1 \over {\sin ^2 x}} = - \csc ^2 x,\,\,\,\,d\left( {\cot x} \right) = - \csc ^2 x \cdot dx y=cotx,y′=−sin2x1=−csc2x,d(cotx)=−csc2x⋅dx
y = sec x , y ′ = sec x tan x , d ( sec x ) = sec x tan x ⋅ d x y = \sec x,\,\,\,\,y' = \sec x\tan x,\,\,\,\,d\left( {\sec x} \right) = \sec x\tan x \cdot dx y=secx,y′=secxtanx,d(secx)=secxtanx⋅dx
y = csc x , y ′ = − csc ⋅ cot x , d ( csc x ) = − csc x ⋅ cot x ⋅ d x y = \csc x,\,\,\,\,y' = - \csc \cdot \cot x,\,\,\,\,d\left( {\csc x} \right) = - \csc x \cdot \cot x \cdot dx y=cscx,y′=−csc⋅cotx,d(cscx)=−cscx⋅cotx⋅dx
y = arcsin x , y ′ = 1 1 − x 2 , d ( arcsin x ) = 1 1 − x 2 ⋅ d x y = \arcsin x,\,\,\,\,y' = {1 \over {\sqrt {1 - x^2 } }},\,\,\,\,d\left( {\arcsin x} \right) = {1 \over {\sqrt {1 - x^2 } }} \cdot dx y=arcsinx,y′=1−x21,d(arcsinx)=1−x21⋅dx
y = arccos x , y ′ = − 1 1 − x 2 , d ( arccos x ) = − 1 1 − x 2 ⋅ d x y = \arccos x,\,\,\,\,y' = - {1 \over {\sqrt {1 - x^2 } }},\,\,\,\,d\left( {\arccos x} \right) = - {1 \over {\sqrt {1 - x^2 } }} \cdot dx y=arccosx,y′=−1−x21,d(arccosx)=−1−x21⋅dx
y = arctan x , y ′ = 1 1 + x 2 , d ( arctan x ) = 1 1 + x 2 ⋅ d x y = \arctan x,\,\,\,\,y' = {1 \over {1 + x^2 }},\,\,\,\,d\left( {\arctan x} \right) = {1 \over {1 + x^2 }} \cdot dx y=arctanx,y′=1+x21,d(arctanx)=1+x21⋅dx
y = a r c cot x , y ′ = − 1 1 + x 2 , d ( a r c cot x ) = − 1 1 + x 2 ⋅ d x y = arc\cot x,\,\,\,\,y' = - {1 \over {1 + x^2 }},\,\,\,\,d\left( {arc\cot x} \right) = - {1 \over {1 + x^2 }} \cdot dx y=arccotx,y′=−1+x21,d(arccotx)=−1+x21⋅dx
y = s h ( x ) = e x − e − x 2 , y ′ = c h ( x ) , d [ s h ( x ) ] = c h ( x ) ⋅ d x y = sh\left( x \right) = {{e^x - e^{ - x} } \over 2},\,\,\,\,y' = ch\left( x \right),\,\,\,\,d\left[ {sh\left( x \right)} \right] = ch\left( x \right) \cdot dx y=sh(x)=2ex−e−x,y′=ch(x),d[sh(x)]=ch(x)⋅dx
y = c h ( x ) = e x + e − x 2 , y ′ = s h ( x ) , d [ c h ( x ) ] = s h ( x ) ⋅ d x y = ch\left( x \right) = {{e^x + e^{ - x} } \over 2},\,\,\,\,y' = sh\left( x \right),\,\,\,\,d\left[ {ch\left( x \right)} \right] = sh\left( x \right) \cdot dx y=ch(x)=2ex+e−x,y′=sh(x),d[ch(x)]=sh(x)⋅dx
y = tanh ( x ) = e x − e − x e x + e − x , y ′ = 1 − tanh 2 ( x ) , d [ tanh ( x ) ] = [ 1 − tanh 2 ( x ) ] ⋅ d x y = \tanh \left( x \right) = {{e^x - e^{ - x} } \over {e^x + e^{ - x} }},\,\,\,\,y' = 1 - \tanh ^2 \left( x \right),\,\,\,\,d\left[ {\tanh \left( x \right)} \right] = \left[ {1 - \tanh ^2 \left( x \right)} \right] \cdot dx y=tanh(x)=ex+e−xex−e−x,y′=1−tanh2(x),d[tanh(x)]=[1−tanh2(x)]⋅dx
y = s i g m o i d ( x ) = 1 1 + e − x = 1 2 [ 1 + tanh ( x ) ] , y ′ = ( 1 − y ) ⋅ y = e − x ( 1 + e − x ) 2 y = sigmoid\left( x \right) = {1 \over {1 + e^{ - x} }} = {1 \over 2}\left[ {1 + \tanh \left( x \right)} \right],\,\,\,\,y' = \left( {1 - y} \right) \cdot y = {{e^{ - x} } \over {\left( {1 + e^{ - x} } \right)^2 }} y=sigmoid(x)=1+e−x1=21[1+tanh(x)],y′=(1−y)⋅y=(1+e−x)2e−x
1.3 隐函数
1.3.1 常用高阶导数公式
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\left( {a^x } \right)^{\left( n \right)} = a^x \ln ^n a,\,\,\,\,\left( {e^x } \right)^{\left( n \right)} = e^x
(ax)(n)=axlnna,(ex)(n)=ex
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(sinkx)(n)=kn⋅(kx+n⋅2π)
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\left( {\cos kx} \right)^{\left( n \right)} = k^n \cos \left( {kx + n \cdot {\pi \over 2}} \right)
(coskx)(n)=kncos(kx+n⋅2π)
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\left( {x^m } \right)^{\left( n \right)} = m\left( {m - 1} \right) \cdots \left( {m - n + 1} \right) \cdot x^{m - n}
(xm)(n)=m(m−1)⋯(m−n+1)⋅xm−n
( ln x ) ( n ) = ( − 1 ) ( n − 1 ) ( n − 1 ) ! x n \left( {\ln x} \right)^{\left( n \right)} = \left( { - 1} \right)^{\left( {n - 1} \right)} {{\left( {n - 1} \right)!} \over {x^n }} (lnx)(n)=(−1)(n−1)xn(n−1)!
莱布尼兹公式:若 u ( x ) , v ( x ) u(x)\,,v(x) u(x),v(x)均 n n n阶可导,则
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{{(uv)}^{(n)}}=\sum\limits_{i={0}}^{n}{c_{n}^{i}{{u}^{(i)}}{{v}^{(n-i)}}}
(uv)(n)=i=0∑ncniu(i)v(n−i)
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1.3.2 反函数的运算法则
设 y = f ( x ) y=f(x) y=f(x)在点 x x x的某邻域内单调连续,在点 x x x处可导且 f ′ ( x ) ≠ 0 {f}'(x)\ne 0 f′(x)=0,则其反函数在点 x x x所对应的 y y y处可导,并且有 d y d x = 1 d x d y \frac{dy}{dx}=\frac{1}{\frac{dx}{dy}} dxdy=dydx1
1.3.3 复合函数求导运算法则:
若 μ = φ ( x ) \mu =\varphi(x) μ=φ(x) 在点 x x x可导,而 y = f ( μ ) y=f(\mu) y=f(μ)在对应点 μ \mu μ( μ = φ ( x ) \mu =\varphi (x) μ=φ(x))可导,则复合函数 y = f ( φ ( x ) ) y=f(\varphi (x)) y=f(φ(x))在点 x x x可导,且 y ′ = f ′ ( μ ) ⋅ φ ′ ( x ) {y}'={f}'(\mu )\cdot {\varphi }'(x) y′=f′(μ)⋅φ′(x)
1.3.4 隐函数求导法则
隐函数导数 d y d x \frac{dy}{dx} dxdy的求法一般有三种方法:
(1)方程两边同时求导
方程两边对 x x x求导,要记住 y y y是 x x x的函数,则 y y y的函数是 x x x的复合函数。
例如 1 y \frac{1}{y} y1, y 2 {{y}^{2}} y2, l n y ln y lny, e y {{{e}}^{y}} ey等均是 x x x的复合函数。对 x x x求导应按复合函数连锁法则做.
(2)公式法
由 F ( x , y ) = 0 F(x,y)=0 F(x,y)=0 知 d y d x = − F ′ x ( x , y ) F ′ y ( x , y ) \frac{dy}{dx}=-\frac{{{{{F}'}}_{x}}(x,y)}{{{{{F}'}}_{y}}(x,y)} dxdy=−F′y(x,y)F′x(x,y)。
其中, F ′ x ( x , y ) {{{F}'}_{x}}(x,y) F′x(x,y), F ′ y ( x , y ) {{{F}'}_{y}}(x,y) F′y(x,y)分别表示 F ( x , y ) F(x,y) F(x,y)对 x x x和 y y y的偏导数
(3)利用微分形式不变性
1.4 法则
1.4.1 费马定理
若函数 f ( x ) f(x) f(x)满足条件:
- 函数
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f ( x ) ≤ f ( x 0 ) f(x)\le f({{x}_{0}}) f(x)≤f(x0)或 f ( x ) ≥ f ( x 0 ) f(x)\ge f({{x}_{0}}) f(x)≥f(x0), - f ( x ) f(x) f(x)在 x 0 {{x}_{0}} x0处可导,则有 f ′ ( x 0 ) = 0 {f}'({{x}_{0}})=0 f′(x0)=0
1.4.2 罗尔定理
设函数 f ( x ) f(x) f(x)满足条件:
- 在闭区间 [ a , b ] [a,b] [a,b]上连续;
- 在 ( a , b ) (a,b) (a,b)内可导;
- f ( a ) = f ( b ) f(a)=f(b) f(a)=f(b);
则在 ( a , b ) (a,b) (a,b)内一存在个 ξ \xi ξ,使 f ′ ( ξ ) = 0 {f}'(\xi )=0 f′(ξ)=0
1.4.3 拉格朗日中值定理
设函数 f ( x ) f(x) f(x)满足条件:
- 在 [ a , b ] [a,b] [a,b]上连续;
- 在 ( a , b ) (a,b) (a,b)内可导;
则在 ( a , b ) (a,b) (a,b)内一存在个 ξ \xi ξ,使 f ( b ) − f ( a ) b − a = f ′ ( ξ ) \frac{f(b)-f(a)}{b-a}={f}'(\xi ) b−af(b)−f(a)=f′(ξ)
1.4.4 柯西中值定理
设函数 f ( x ) f(x) f(x), g ( x ) g(x) g(x)满足条件:
- 在 [ a , b ] [a,b] [a,b]上连续;
- 在 ( a , b ) (a,b) (a,b)内可导且 f ′ ( x ) {f}'(x) f′(x), g ′ ( x ) {g}'(x) g′(x)均存在,且 g ′ ( x ) ≠ 0 {g}'(x)\ne 0 g′(x)=0
则在 ( a , b ) (a,b) (a,b)内存在一个 ξ \xi ξ,使 f ( b ) − f ( a ) g ( b ) − g ( a ) = f ′ ( ξ ) g ′ ( ξ ) \frac{f(b)-f(a)}{g(b)-g(a)}=\frac{{f}'(\xi )}{{g}'(\xi )} g(b)−g(a)f(b)−f(a)=g′(ξ)f′(ξ)
1.4.5 洛必达法则
(1)
设函数
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\mathop {\lim }\limits_{x \to x_0 } f\left( x \right) = 0,\,\,\,\,\mathop {\lim }\limits_{x \to x_0 } g\left( x \right) = 0
x→x0limf(x)=0,x→x0limg(x)=0
f ( x ) , g ( x ) f\left( x \right),g\left( x \right) f(x),g(x)在 x 0 {{x}_{0}} x0的邻域内可导,(在 x 0 {{x}_{0}} x0处可除外),且 g ′ ( x ) ≠ 0 {g}'\left( x \right)\ne 0 g′(x)=0; lim x → x 0 f ′ ( x ) g ′ ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)} x→x0limg′(x)f′(x)存在(或 ∞ \infty ∞)。则:
lim x → x 0 f ( x ) g ( x ) = lim x → x 0 f ′ ( x ) g ′ ( x ) \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)} x→x0limg(x)f(x)=x→x0limg′(x)f′(x)
(2)
设函数 f ( x ) , g ( x ) f\left( x \right),g\left( x \right) f(x),g(x)满足条件:
lim x → ∞ f ( x ) = 0 , lim x → ∞ g ( x ) = 0 \underset{x\to \infty }{\mathop{\lim }}\,f\left( x \right)=0,\underset{x\to \infty }{\mathop{\lim }}\,g\left( x \right)=0 x→∞limf(x)=0,x→∞limg(x)=0
存在一个
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\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)}
x→x0limg(x)f(x)=x→x0limg′(x)f′(x)
(3)
设函数
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\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,f\left( x \right)=\infty,\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,g\left( x \right)=\infty
x→x0limf(x)=∞,x→x0limg(x)=∞
f ( x ) , g ( x ) f\left( x \right),g\left( x \right) f(x),g(x) 在 x 0 {{x}_{0}} x0 的邻域内可导(在 x 0 {{x}_{0}} x0处可除外)且 g ′ ( x ) ≠ 0 {g}'\left( x \right)\ne 0 g′(x)=0; lim x → x 0 f ′ ( x ) g ′ ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)} x→x0limg′(x)f′(x) 存在(或 ∞ \infty ∞)。则:
lim x → x 0 f ( x ) g ( x ) = lim x → x 0 f ′ ( x ) g ′ ( x ) \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)} x→x0limg(x)f(x)=x→x0limg′(x)f′(x) 同理法则 I I ′ {I{I}'} II′ ( ∞ ∞ \frac{\infty }{\infty } ∞∞ 型)仿法则 I ′ {{I}'} I′ 可写出。
1.4.6 泰勒公式
设函数 f ( x ) f(x) f(x)在点 x 0 {{x}_{0}} x0处的某邻域内具有 n + 1 n+1 n+1阶导数,则对该邻域内异于 x 0 {{x}_{0}} x0的任意点 x x x,在 x 0 {{x}_{0}} x0与 x x x之间至少存在一个 ξ \xi ξ,使得:
f ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + 1 2 ! f ′ ′ ( x 0 ) ( x − x 0 ) 2 + ⋯ + f ( n ) ( x 0 ) n ! ( x − x 0 ) n + R n ( x ) f(x)=f({{x}_{0}})+{f}'({{x}_{0}})(x-{{x}_{0}})+\frac{1}{2!}{f}''({{x}_{0}}){{(x-{{x}_{0}})}^{2}}+\cdots+\frac{{{f}^{(n)}}({{x}_{0}})}{n!}{{(x-{{x}_{0}})}^{n}}+{{R}_{n}}(x) f(x)=f(x0)+f′(x0)(x−x0)+2!1f′′(x0)(x−x0)2+⋯+n!f(n)(x0)(x−x0)n+Rn(x)
其中 R n ( x ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! ( x − x 0 ) n + 1 {{R}_{n}}(x)=\frac{{{f}^{(n+1)}}(\xi )}{(n+1)!}{{(x-{{x}_{0}})}^{n+1}} Rn(x)=(n+1)!f(n+1)(ξ)(x−x0)n+1称为 f ( x ) f(x) f(x)在点 x 0 {{x}_{0}} x0处的 n n n阶泰勒余项。
令 x 0 = 0 {{x}_{0}}=0 x0=0,则 n n n阶泰勒公式
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Rn(x)=(n+1)!f(n+1)(ξ)xn+1,$\xi
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在0与x$之间.(1)式称为麦克劳林公式
(1)五种函数的台劳级数
Ⅰ.
e x = 1 + x + 1 2 ! x 2 + ⋯ + 1 n ! x n + x n + 1 ( n + 1 ) ! e ξ {{{e}}^{x}}=1+x+\frac{1}{2!}{{x}^{2}}+\cdots +\frac{1}{n!}{{x}^{n}}+\frac{{{x}^{n+1}}}{(n+1)!}{{e}^{\xi }} ex=1+x+2!1x2+⋯+n!1xn+(n+1)!xn+1eξ
或 = 1 + x + 1 2 ! x 2 + ⋯ + 1 n ! x n + o ( x n ) =1+x+\frac{1}{2!}{{x}^{2}}+\cdots +\frac{1}{n!}{{x}^{n}}+o({{x}^{n}}) =1+x+2!1x2+⋯+n!1xn+o(xn)
Ⅱ.
sin x = x − 1 3 ! x 3 + ⋯ + x n n ! sin n π 2 + x n + 1 ( n + 1 ) ! sin ( ξ + n + 1 2 π ) \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 ) sinx=x−3!1x3+⋯+n!xnsin2nπ+(n+1)!xn+1sin(ξ+2n+1π)
或 = x − 1 3 ! x 3 + ⋯ + x n n ! sin n π 2 + o ( x n ) =x-\frac{1}{3!}{{x}^{3}}+\cdots +\frac{{{x}^{n}}}{n!}\sin \frac{n\pi }{2}+o({{x}^{n}}) =x−3!1x3+⋯+n!xnsin2nπ+o(xn)
Ⅲ.
cos x = 1 − 1 2 ! x 2 + ⋯ + x n n ! cos n π 2 + x n + 1 ( n + 1 ) ! cos ( ξ + n + 1 2 π ) \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 ) cosx=1−2!1x2+⋯+n!xncos2nπ+(n+1)!xn+1cos(ξ+2n+1π)
或 = 1 − 1 2 ! x 2 + ⋯ + x n n ! cos n π 2 + o ( x n ) =1-\frac{1}{2!}{{x}^{2}}+\cdots +\frac{{{x}^{n}}}{n!}\cos \frac{n\pi }{2}+o({{x}^{n}}) =1−2!1x2+⋯+n!xncos2nπ+o(xn)
Ⅳ.
ln ( 1 + x ) = x − 1 2 x 2 + 1 3 x 3 − ⋯ + ( − 1 ) n − 1 x n n + ( − 1 ) n x n + 1 ( n + 1 ) ( 1 + ξ ) n + 1 \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}}} ln(1+x)=x−21x2+31x3−⋯+(−1)n−1nxn+(n+1)(1+ξ)n+1(−1)nxn+1
或 = x − 1 2 x 2 + 1 3 x 3 − ⋯ + ( − 1 ) n − 1 x n n + o ( x n ) =x-\frac{1}{2}{{x}^{2}}+\frac{1}{3}{{x}^{3}}-\cdots +{{(-1)}^{n-1}}\frac{{{x}^{n}}}{n}+o({{x}^{n}}) =x−21x2+31x3−⋯+(−1)n−1nxn+o(xn)
Ⅴ.m
( 1 + x ) m = 1 + m x + m ( m − 1 ) 2 ! x 2 + ⋯ + m ( m − 1 ) ⋯ ( m − n + 1 ) n ! x n + m ( m − 1 ) ⋯ ( m − n + 1 ) ( n + 1 ) ! x n + 1 ( 1 + ξ ) m − n − 1 {{(1+x)}^{m}}=1+mx+\frac{m(m-1)}{2!}{{x}^{2}}+\cdots +\frac{m(m-1)\cdots (m-n+1)}{n!}{{x}^{n}}+\frac{m(m-1)\cdots (m-n+1)}{(n+1)!}{{x}^{n+1}}{{(1+\xi )}^{m-n-1}} (1+x)m=1+mx+2!m(m−1)x2+⋯+n!m(m−1)⋯(m−n+1)xn+(n+1)!m(m−1)⋯(m−n+1)xn+1(1+ξ)m−n−1
或 ( 1 + x ) m = 1 + m x + m ( m − 1 ) 2 ! x 2 + ⋯ + m ( m − 1 ) ⋯ ( m − n + 1 ) n ! x n + o ( x n ) {{(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}}) (1+x)m=1+mx+2!m(m−1)x2+⋯+n!m(m−1)⋯(m−n+1)xn+o(xn)
1.5 函数单调性
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Th1: 设函数 f ( x ) f(x) f(x)在 ( a , b ) (a,b) (a,b)区间内可导,如果对 ∀ x ∈ ( a , b ) \forall x\in (a,b) ∀x∈(a,b),都有 f ′ ( x ) > 0 f\,'(x)>0 f′(x)>0(或 f ′ ( x ) < 0 f\,'(x)<0 f′(x)<0),则函数 f ( x ) f(x) f(x)在 ( a , b ) (a,b) (a,b)内是单调增加的(或单调减少)
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Th2: (取极值的必要条件)设函数 f ( x ) f(x) f(x)在 x 0 {{x}_{0}} x0处可导,且在 x 0 {{x}_{0}} x0处取极值,则 f ′ ( x 0 ) = 0 f\,'({{x}_{0}})=0 f′(x0)=0。
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Th3: (取极值的第一充分条件)设函数 f ( x ) f(x) f(x)在 x 0 {{x}_{0}} x0的某一邻域内可微,且 f ′ ( x 0 ) = 0 f\,'({{x}_{0}})=0 f′(x0)=0(或 f ( x ) f(x) f(x)在 x 0 {{x}_{0}} x0处连续,但 f ′ ( x 0 ) f\,'({{x}_{0}}) f′(x0)不存在。)
- 若当 x x x经过 x 0 {{x}_{0}} x0时, f ′ ( x ) f\,'(x) f′(x)由“+”变“-”,则 f ( x 0 ) f({{x}_{0}}) f(x0)为极大值;
- 若当 x x x经过 x 0 {{x}_{0}} x0时, f ′ ( x ) f\,'(x) f′(x)由“-”变“+”,则 f ( x 0 ) f({{x}_{0}}) f(x0)为极小值;
- 若 f ′ ( x ) f\,'(x) f′(x)经过 x = x 0 x={{x}_{0}} x=x0的两侧不变号,则 f ( x 0 ) f({{x}_{0}}) f(x0)不是极值。
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Th4: (取极值的第二充分条件)设 f ( x ) f(x) f(x)在点 x 0 {{x}_{0}} x0处有 f ′ ′ ( x ) ≠ 0 f''(x)\ne 0 f′′(x)=0,且 f ′ ( x 0 ) = 0 f\,'({{x}_{0}})=0 f′(x0)=0,则 当 f ′ ′ ( x 0 ) < 0 f'\,'({{x}_{0}})<0 f′′(x0)<0时, f ( x 0 ) f({{x}_{0}}) f(x0)为极大值;当 f ′ ′ ( x 0 ) > 0 f'\,'({{x}_{0}})>0 f′′(x0)>0时, f ( x 0 ) f({{x}_{0}}) f(x0)为极小值。
注:如果 f ′ ′ ( x 0 ) < 0 f'\,'({{x}_{0}})<0 f′′(x0)<0,此方法失效。
1.6 函数渐近线
1.6.1 水平渐近线
若 lim x → + ∞ f ( x ) = b \underset{x\to +\infty }{\mathop{\lim }}\,f(x)=b x→+∞limf(x)=b,或 lim x → − ∞ f ( x ) = b \underset{x\to -\infty }{\mathop{\lim }}\,f(x)=b x→−∞limf(x)=b,则
y = b y=b y=b称为函数 y = f ( x ) y=f(x) y=f(x)的水平渐近线。
1.6.2 垂直渐近线
若 lim x → x 0 − f ( x ) = ∞ \underset{x\to x_{0}^{-}}{\mathop{\lim }}\,f(x)=\infty x→x0−limf(x)=∞,或 lim x → x 0 + f ( x ) = ∞ \underset{x\to x_{0}^{+}}{\mathop{\lim }}\,f(x)=\infty x→x0+limf(x)=∞,则
x = x 0 x={{x}_{0}} x=x0称为 y = f ( x ) y=f(x) y=f(x)的铅直渐近线。
1.6.3 倾斜渐近线
若 a = lim x → ∞ f ( x ) x , b = lim x → ∞ [ f ( x ) − a x ] a=\underset{x\to \infty }{\mathop{\lim }}\,\frac{f(x)}{x},\quad b=\underset{x\to \infty }{\mathop{\lim }}\,[f(x)-ax] a=x→∞limxf(x),b=x→∞lim[f(x)−ax],则 y = a x + b y=ax+b y=ax+b称为 y = f ( x ) y=f(x) y=f(x)的斜渐近线。
1.7 函数曲线相关特性
1.7.1 函数凸凹性
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Th1: (凹凸性的判别定理)若在I上 f ′ ′ ( x ) < 0 f''(x)<0 f′′(x)<0(或 f ′ ′ ( x ) > 0 f''(x)>0 f′′(x)>0),则 f ( x ) f(x) f(x)在I上是凸的(或凹的)。
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Th2: (拐点的判别定理1)若在 x 0 {{x}_{0}} x0处 f ′ ′ ( x ) = 0 f''(x)=0 f′′(x)=0,(或 f ′ ′ ( x ) f''(x) f′′(x)不存在),当 x x x变动经过 x 0 {{x}_{0}} x0时, f ′ ′ ( x ) f''(x) f′′(x)变号,则 ( x 0 , f ( x 0 ) ) ({{x}_{0}},f({{x}_{0}})) (x0,f(x0))为拐点。
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Th3: (拐点的判别定理2)设 f ( x ) f(x) f(x)在 x 0 {{x}_{0}} x0点的某邻域内有三阶导数,且 f ′ ′ ( x ) = 0 f''(x)=0 f′′(x)=0, f ′ ′ ′ ( x ) ≠ 0 f'''(x)\ne 0 f′′′(x)=0,则 ( x 0 , f ( x 0 ) ) ({{x}_{0}},f({{x}_{0}})) (x0,f(x0))为拐点。
1.7.2 弧微分
d S = 1 + y ′ 2 d x dS=\sqrt{1+y{{'}^{2}}}dx dS=1+y′2dx
1.7.3 曲率
曲线 y = f ( x ) y=f(x) y=f(x)在点 ( x , y ) (x,y) (x,y)处的曲率
k = ∣ y ′ ′ ∣ ( 1 + y ′ 2 ) 3 2 k=\frac{\left| y'' \right|}{{{(1+y{{'}^{2}})}^{\tfrac{3}{2}}}} k=(1+y′2)23∣y′′∣
对于参数方程 x = ϕ ( t ) , y = ψ ( t ) x = \phi \left( t \right),y = \psi \left( t \right) x=ϕ(t),y=ψ(t):
k = ∣ φ ′ ( t ) ψ ′ ′ ( t ) − φ ′ ′ ( t ) ψ ′ ( t ) ∣ [ φ ′ 2 ( t ) + ψ ′ 2 ( t ) ] 3 2 k=\frac{\left| \varphi '(t)\psi ''(t)-\varphi ''(t)\psi '(t) \right|}{{{[\varphi {{'}^{2}}(t)+\psi {{'}^{2}}(t)]}^{\tfrac{3}{2}}}} k=[φ′2(t)+ψ′2(t)]23∣φ′(t)ψ′′(t)−φ′′(t)ψ′(t)∣
1.7.4 曲率半径
曲线在点 M M M处的曲率 k ( k ≠ 0 ) k(k\ne 0) k(k=0)与曲线在点 M M M处的曲率半径 ρ \rho ρ有如下关系: ρ = 1 k \rho =\frac{1}{k} ρ=k1
§02 线性代数
2.1 行列式
2.1.1 行列式展开定理
设 A = ( a i j ) n × n A = ( a_{{ij}} )_{n \times n} A=(aij)n×n,则: a i 1 A j 1 + a i 2 A j 2 + ⋯ + a i n A j n = { ∣ A ∣ , i = j 0 , i ≠ j a_{i1}A_{j1} +a_{i2}A_{j2} + \cdots + a_{{in}}A_{{jn}} = \begin{cases}|A|,i=j\\ 0,i \neq j\end{cases} ai1Aj1+ai2Aj2+⋯+ainAjn={∣A∣,i=j0,i=j
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a_{1i}A_{1j} + a_{2i}A_{2j} + \cdots + a_{{ni}}A_{{nj}} = \begin{cases}|A|,i=j\\ 0,i \neq j\end{cases}
a1iA1j+a2iA2j+⋯+aniAnj={∣A∣,i=j0,i=j
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其中: A ∗ = ( A 11 A 12 … A 1 n A 21 A 22 … A 2 n … … … … A n 1 A n 2 … A n n ) = ( A j i ) = ( A i j ) T 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} A∗=⎝⎜⎜⎛A11A21…An1A12A22…An2…………A1nA2n…Ann⎠⎟⎟⎞=(Aji)=(Aij)T
2.1.2 转置行列式
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设 A , B A,B A,B为 n n n阶方阵,则 ∣ A B ∣ = ∣ A ∣ ∣ B ∣ = ∣ B ∣ ∣ A ∣ = ∣ B A ∣ \left| {AB} \right| = \left| A \right|\left| B \right| = \left| B \right|\left| A \right| = \left| {BA} \right| ∣AB∣=∣A∣∣B∣=∣B∣∣A∣=∣BA∣
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∣ A ± B ∣ = ∣ A ∣ ± ∣ B ∣ \left| A \pm B \right| = \left| A \right| \pm \left| B \right| ∣A±B∣=∣A∣±∣B∣不一定成立。
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∣ k A ∣ = k n ∣ A ∣ \left| {kA} \right| = k^{n}\left| A \right| ∣kA∣=kn∣A∣, A A A为 n n n阶方阵。
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设 A A A为 n n n阶方阵, ∣ A T ∣ = ∣ A ∣ ; ∣ A − 1 ∣ = ∣ A ∣ − 1 |A^{T}| = |A|;|A^{- 1}| = |A|^{- 1} ∣AT∣=∣A∣;∣A−1∣=∣A∣−1(若 A A A可逆), ∣ A ∗ ∣ = ∣ A ∣ n − 1 |A^{*}| = |A|^{n - 1} ∣A∗∣=∣A∣n−1, n ≥ 2 n \geq 2 n≥2
2.1.3 对角线方阵行列式
∣ A O O B ∣ = ∣ A C O B ∣ = ∣ A O C B ∣ = ∣ A ∣ ∣ B ∣ \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| ∣∣∣∣AOOB∣∣∣∣=∣∣∣∣ACOB∣∣∣∣=∣∣∣∣AOCB∣∣∣∣=∣A∣∣B∣
A , B A,B A,B为方阵
∣ O A m × m B n × n O ∣ = ( − 1 ) m n ∣ A ∣ ∣ B ∣ \left| \begin{matrix} {O} & A_{m \times m} \\ B_{n \times n} & { O} \\ \end{matrix} \right| = ({- 1)}^{{mn}}|A||B| ∣∣∣∣OBn×nAm×mO∣∣∣∣=(−1)mn∣A∣∣B∣
2.1.4 范德蒙德矩阵
范德蒙行列式 D n = ∣ 1 1 … 1 x 1 x 2 … x n … … … … x 1 n − 1 x 2 n 1 … x n n − 1 ∣ = ∏ 1 ≤ j < i ≤ n ( x i − x j ) 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}) Dn=∣∣∣∣∣∣∣∣1x1…x1n−11x2…x2n1…………1xn…xnn−1∣∣∣∣∣∣∣∣=1≤j<i≤n∏(xi−xj)
2.1.5 特征值
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2.2 矩阵
2.2.1 定义
矩阵: m × n m \times n m×n个数 a i j a_{{ij}} aij排成 m m m行 n n n列的表格 [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋯ ⋯ ⋯ ⋯ ⋯ a m 1 a m 2 ⋯ a m n ] \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} ⎣⎢⎢⎡a11a12⋯a1na21a22⋯a2n⋯⋯⋯⋯⋯am1am2⋯amn⎦⎥⎥⎤ 称为矩阵,简记为 A A A,或者 ( a i j ) m × n \left( a_{{ij}} \right)_{m \times n} (aij)m×n 。若 m = n m = n m=n,则称 A A A是 n n n阶矩阵或 n n n阶方阵。
2.2.2 矩阵线性运算
(1)矩阵加法
设 A = ( a i j ) , B = ( b i j ) A = (a_{{ij}}),B = (b_{{ij}}) A=(aij),B=(bij)是两个 m × n m \times n m×n矩阵,则 m × n m \times n m×n 矩阵 C = c i j ) = a i j + b i j C = c_{{ij}}) = a_{{ij}} + b_{{ij}} C=cij)=aij+bij称为矩阵 A A A与 B B B的和,记为 A + B = C A + B = C A+B=C 。
(2)矩阵数乘
设 A = ( a i j ) A = (a_{{ij}}) A=(aij)是 m × n m \times n m×n矩阵, k k k是一个常数,则 m × n m \times n m×n矩阵 ( k a i j ) (ka_{{ij}}) (kaij)称为数 k k k与矩阵 A A A的数乘,记为 k A {kA} kA。
(3)矩阵乘法
设 A = ( a i j ) A = (a_{{ij}}) A=(aij)是 m × n m \times n m×n矩阵, B = ( b i j ) B = (b_{{ij}}) B=(bij)是 n × s n \times s n×s矩阵,那么 m × s m \times s m×s矩阵 C = ( c i j ) C = (c_{{ij}}) C=(cij),其中 c i j = a i 1 b 1 j + a i 2 b 2 j + ⋯ + a i n b n j = ∑ k = 1 n a i k b k j c_{{ij}} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{{in}}b_{{nj}} = \sum_{k =1}^{n}{a_{{ik}}b_{{kj}}} cij=ai1b1j+ai2b2j+⋯+ainbnj=∑k=1naikbkj称为 A B {AB} AB的乘积,记为 C = A B C = AB C=AB 。
(4)A*关系
Ⅰ.AT
( A T ) T = A , ( A B ) T = B T A T , ( k A ) T = k A T , ( A ± B ) T = A T ± B T {(A^{T})}^{T} = A,{(AB)}^{T} = B^{T}A^{T},{(kA)}^{T} = kA^{T},{(A \pm B)}^{T} = A^{T} \pm B^{T} (AT)T=A,(AB)T=BTAT,(kA)T=kAT,(A±B)T=AT±BT
Ⅱ.A-1
( A − 1 ) − 1 = A , ( A B ) − 1 = B − 1 A − 1 , ( k A ) − 1 = 1 k A − 1 \left( A^{- 1} \right)^{- 1} = A,\left( {AB} \right)^{- 1} = B^{- 1}A^{- 1},\left( {kA} \right)^{- 1} = \frac{1}{k}A^{- 1} (A−1)−1=A,(AB)−1=B−1A−1,(kA)−1=k1A−1
但 ( A ± B ) − 1 = A − 1 ± B − 1 {(A \pm B)}^{- 1} = A^{- 1} \pm B^{- 1} (A±B)−1=A−1±B−1不一定成立。
Ⅲ.A*
( A ∗ ) ∗ = ∣ A ∣ n − 2 A ( n ≥ 3 ) \left( A^{*} \right)^{*} = |A|^{n - 2}\ A\ \ (n \geq 3) (A∗)∗=∣A∣n−2 A (n≥3) ( A B ) ∗ = B ∗ A ∗ \left({AB} \right)^{*} = B^{*}A^{*} (AB)∗=B∗A∗ ( k A ) ∗ = k n − 1 A ∗ ( n ≥ 2 ) \left( {kA} \right)^{*} = k^{n -1}A^{*}{\ \ }\left( n \geq 2 \right) (kA)∗=kn−1A∗ (n≥2)
但 ( A ± B ) ∗ = A ∗ ± B ∗ \left( A \pm B \right)^{*} = A^{*} \pm B^{*} (A±B)∗=A∗±B∗不一定成立。
Ⅳ.A-1,T
( A − 1 ) T = ( A T ) − 1 , ( A − 1 ) ∗ = ( A A ∗ ) − 1 , ( A ∗ ) T = ( A T ) ∗ {(A^{- 1})}^{T} = {(A^{T})}^{- 1},\ \left( A^{- 1} \right)^{*} ={(AA^{*})}^{- 1},{(A^{*})}^{T} = \left( A^{T} \right)^{*} (A−1)T=(AT)−1, (A−1)∗=(AA∗)−1,(A∗)T=(AT)∗
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A A ∗ = A ∗ A = ∣ A ∣ E AA^{*} = A^{*}A = |A|E AA∗=A∗A=∣A∣E
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∣ A ∗ ∣ = ∣ A ∣ n − 1 ( n ≥ 2 ) , ( k A ) ∗ = k n − 1 A ∗ , ( A ∗ ) ∗ = ∣ A ∣ n − 2 A ( n ≥ 3 ) |A^{*}| = |A|^{n - 1}\ (n \geq 2),\ \ \ \ {(kA)}^{*} = k^{n -1}A^{*},{{\ \ }\left( A^{*} \right)}^{*} = |A|^{n - 2}A(n \geq 3) ∣A∗∣=∣A∣n−1 (n≥2), (kA)∗=kn−1A∗, (A∗)∗=∣A∣n−2A(n≥3)
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若 A A A可逆,则 A ∗ = ∣ A ∣ A − 1 , ( A ∗ ) ∗ = 1 ∣ A ∣ A A^{*} = |A|A^{- 1},{(A^{*})}^{*} = \frac{1}{|A|}A A∗=∣A∣A−1,(A∗)∗=∣A∣1A
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若 A A A为 n n n阶方阵,则:
r ( A ∗ ) = { n , r ( A ) = n 1 , r ( A ) = n − 1 0 , r ( A ) < n − 1 r(A^*)=\begin{cases}n,\quad r(A)=n\\ 1,\quad r(A)=n-1\\ 0,\quad r(A)<n-1\end{cases} r(A∗)=⎩⎪⎨⎪⎧n,r(A)=n1,r(A)=n−10,r(A)<n−1
2.2.3 有关A-1相关结论
A A A可逆 ⇔ A B = E ; ⇔ ∣ A ∣ ≠ 0 ; ⇔ r ( A ) = n ; \Leftrightarrow AB = E; \Leftrightarrow |A| \neq 0; \Leftrightarrow r(A) = n; ⇔AB=E;⇔∣A∣=0;⇔r(A)=n;
⇔ A \Leftrightarrow A ⇔A可以表示为初等矩阵的乘积; ⇔ A ; ⇔ A x = 0 \Leftrightarrow A;\Leftrightarrow Ax = 0 ⇔A;⇔Ax=0。
2.2.4 有关矩阵秩结论
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秩 r ( A ) r(A) r(A)=行秩=列秩;
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r ( A m × n ) ≤ min ( m , n ) ; r(A_{m \times n}) \leq \min(m,n); r(Am×n)≤min(m,n);
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A ≠ 0 ⇒ r ( A ) ≥ 1 A \neq 0 \Rightarrow r(A) \geq 1 A=0⇒r(A)≥1;
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r ( A ± B ) ≤ r ( A ) + r ( B ) ; r(A \pm B) \leq r(A) + r(B); r(A±B)≤r(A)+r(B);
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初等变换不改变矩阵的秩
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r ( A ) + r ( B ) − n ≤ r ( A B ) ≤ min ( r ( A ) , r ( B ) ) , r(A) + r(B) - n \leq r(AB) \leq \min(r(A),r(B)), r(A)+r(B)−n≤r(AB)≤min(r(A),r(B)),特别若 A B = O AB = O AB=O,则: r ( A ) + r ( B ) ≤ n r(A) + r(B) \leq n r(A)+r(B)≤n
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若 A − 1 A^{- 1} A−1存在 ⇒ r ( A B ) = r ( B ) ; \Rightarrow r(AB) = r(B); ⇒r(AB)=r(B); 若 B − 1 B^{- 1} B−1存在
⇒ r ( A B ) = r ( A ) ; \Rightarrow r(AB) = r(A); ⇒r(AB)=r(A); -
若 r ( A m × n ) = n ⇒ r ( A B ) = r ( B ) ; r(A_{m \times n}) = n \Rightarrow r(AB) = r(B); r(Am×n)=n⇒r(AB)=r(B); 若 r ( A m × s ) = n ⇒ r ( A B ) = r ( A ) r(A_{m \times s}) = n\Rightarrow r(AB) = r\left( A \right) r(Am×s)=n⇒r(AB)=r(A)。
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r ( A m × s ) = n ⇔ A x = 0 r(A_{m \times s}) = n \Leftrightarrow Ax = 0 r(Am×s)=n⇔Ax=0只有零解
2.2.5 分块求逆
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\begin{pmatrix} A & O \\ O & B \\ \end{pmatrix}^{- 1} = \begin{pmatrix} A^{-1} & O \\ O & B^{- 1} \\ \end{pmatrix}
(AOOB)−1=(A−1OOB−1)
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\begin{pmatrix} A & C \\ O & B \\\end{pmatrix}^{- 1} = \begin{pmatrix} A^{- 1}& - A^{- 1}CB^{- 1} \\ O & B^{- 1} \\ \end{pmatrix}
(AOCB)−1=(A−1O−A−1CB−1B−1)
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\begin{pmatrix} A & O \\ C & B \\ \end{pmatrix}^{- 1} = \begin{pmatrix} A^{- 1}&{O} \\ - B^{- 1}CA^{- 1} & B^{- 1} \\\end{pmatrix}
(ACOB)−1=(A−1−B−1CA−1OB−1)
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\begin{pmatrix} O & A \\ B & O \\ \end{pmatrix}^{- 1} =\begin{pmatrix} O & B^{- 1} \\ A^{- 1} & O \\ \end{pmatrix}
(OBAO)−1=(OA−1B−1O)
这里 A A A, B B B均为可逆方阵。
2.3 向量
2.3.1 线性表示
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α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs线性相关 ⇔ \Leftrightarrow ⇔至少有一个向量可以用其余向量线性表示。
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α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs线性无关, α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs, β \beta β线性相关 ⇔ β \Leftrightarrow \beta ⇔β可以由 α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs唯一线性表示。
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β \beta β可以由 α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs线性表示
⇔ r ( α 1 , α 2 , ⋯ , α s ) = r ( α 1 , α 2 , ⋯ , α s , β ) \Leftrightarrow r(\alpha_{1},\alpha_{2},\cdots,\alpha_{s}) =r(\alpha_{1},\alpha_{2},\cdots,\alpha_{s},\beta) ⇔r(α1,α2,⋯,αs)=r(α1,α2,⋯,αs,β) 。
2.3.2 线性相关
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部分相关,整体相关;整体无关,部分无关.
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① n n n个 n n n维向量
α 1 , α 2 ⋯ α n \alpha_{1},\alpha_{2}\cdots\alpha_{n} α1,α2⋯αn线性无关 ⇔ ∣ [ α 1 α 2 ⋯ α n ] ∣ ≠ 0 \Leftrightarrow \left|\left\lbrack \alpha_{1}\alpha_{2}\cdots\alpha_{n} \right\rbrack \right| \neq0 ⇔∣[α1α2⋯αn]∣=0, n n n个 n n n维向量 α 1 , α 2 ⋯ α n \alpha_{1},\alpha_{2}\cdots\alpha_{n} α1,α2⋯αn线性相关
⇔ ∣ [ α 1 , α 2 , ⋯ , α n ] ∣ = 0 \Leftrightarrow |\lbrack\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\rbrack| = 0 ⇔∣[α1,α2,⋯,αn]∣=0
② n + 1 n + 1 n+1个 n n n维向量线性相关。
③ 若 α 1 , α 2 ⋯ α S \alpha_{1},\alpha_{2}\cdots\alpha_{S} α1,α2⋯αS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。
2.3.3 向量组的秩
设 r ( A m × n ) = r r(A_{m \times n}) =r r(Am×n)=r,则 A A A的秩 r ( A ) r(A) r(A)与 A A A的行列向量组的线性相关性关系为:
(1) 若 r ( A m × n ) = r = m r(A_{m \times n}) = r = m r(Am×n)=r=m,则 A A A的行向量组线性无关。
(2) 若 r ( A m × n ) = r < m r(A_{m \times n}) = r < m r(Am×n)=r<m,则 A A A的行向量组线性相关。
(3) 若 r ( A m × n ) = r = n r(A_{m \times n}) = r = n r(Am×n)=r=n,则 A A A的列向量组线性无关。
(4) 若 r ( A m × n ) = r < n r(A_{m \times n}) = r < n r(Am×n)=r<n,则 A A A的列向量组线性相关。
2.3.4 基变换与过渡矩阵
若 α 1 , α 2 , ⋯ , α n \alpha_{1},\alpha_{2},\cdots,\alpha_{n} α1,α2,⋯,αn与 β 1 , β 2 , ⋯ , β n \beta_{1},\beta_{2},\cdots,\beta_{n} β1,β2,⋯,βn是向量空间 V V V的两组基,则基变换公式为:
( β 1 , β 2 , ⋯ , β n ) = ( α 1 , α 2 , ⋯ , α n ) [ c 11 c 12 ⋯ c 1 n c 21 c 22 ⋯ c 2 n ⋯ ⋯ ⋯ ⋯ c n 1 c n 2 ⋯ c n n ] = ( α 1 , α 2 , ⋯ , α n ) C (\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 (β1,β2,⋯,βn)=(α1,α2,⋯,αn)⎣⎢⎢⎡c11c21⋯cn1c12c22⋯cn2⋯⋯⋯⋯c1nc2n⋯cnn⎦⎥⎥⎤=(α1,α2,⋯,αn)C
其中 C C C是可逆矩阵,称为由基 α 1 , α 2 , ⋯ , α n \alpha_{1},\alpha_{2},\cdots,\alpha_{n} α1,α2,⋯,αn到基 β 1 , β 2 , ⋯ , β n \beta_{1},\beta_{2},\cdots,\beta_{n} β1,β2,⋯,βn的过渡矩阵。
2.3.5 坐标变换公式
若向量
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γ在基
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α1,α2,⋯,αn与基
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β1,β2,⋯,βn的坐标分别是
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X=(x1,x2,⋯,xn)T
Y = ( y 1 , y 2 , ⋯ , y n ) T Y = \left( y_{1},y_{2},\cdots,y_{n} \right)^{T} Y=(y1,y2,⋯,yn)T
即: γ = x 1 α 1 + x 2 α 2 + ⋯ + x n α n = y 1 β 1 + y 2 β 2 + ⋯ + y n β n \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} γ=x1α1+x2α2+⋯+xnαn=y1β1+y2β2+⋯+ynβn
则向量坐标变换公式为 X = C Y X = CY X=CY 或 Y = C − 1 X Y = C^{- 1}X Y=C−1X,其中 C C C是从基 α 1 , α 2 , ⋯ , α n \alpha_{1},\alpha_{2},\cdots,\alpha_{n} α1,α2,⋯,αn到基 β 1 , β 2 , ⋯ , β n \beta_{1},\beta_{2},\cdots,\beta_{n} β1,β2,⋯,βn的过渡矩阵。
2.3.6 向量内积
( α , β ) = a 1 b 1 + a 2 b 2 + ⋯ + a n b n = α T β = β T α (\alpha,\beta) = a_{1}b_{1} + a_{2}b_{2} + \cdots + a_{n}b_{n} = \alpha^{T}\beta = \beta^{T}\alpha (α,β)=a1b1+a2b2+⋯+anbn=αTβ=βTα
2.3.7 Schmidt正交化
(1)正交化过程
若 α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs线性无关,则可构造 β 1 , β 2 , ⋯ , β s \beta_{1},\beta_{2},\cdots,\beta_{s} β1,β2,⋯,βs使其两两正交,且 β i \beta_{i} βi仅是 α 1 , α 2 , ⋯ , α i \alpha_{1},\alpha_{2},\cdots,\alpha_{i} α1,α2,⋯,αi的线性组合 ( i = 1 , 2 , ⋯ , n ) (i= 1,2,\cdots,n) (i=1,2,⋯,n),再把 β i \beta_{i} βi单位化,记 γ i = β i ∣ β i ∣ \gamma_{i} =\frac{\beta_{i}}{\left| \beta_{i}\right|} γi=∣βi∣βi,则 γ 1 , γ 2 , ⋯ , γ i \gamma_{1},\gamma_{2},\cdots,\gamma_{i} γ1,γ2,⋯,γi是规范正交向量组。其中
β 1 = α 1 \beta_{1} = \alpha_{1} β1=α1, β 2 = α 2 − ( α 2 , β 1 ) ( β 1 , β 1 ) β 1 \beta_{2} = \alpha_{2} -\frac{(\alpha_{2},\beta_{1})}{(\beta_{1},\beta_{1})}\beta_{1} β2=α2−(β1,β1)(α2,β1)β1 , β 3 = α 3 − ( α 3 , β 1 ) ( β 1 , β 1 ) β 1 − ( α 3 , β 2 ) ( β 2 , β 2 ) β 2 \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} β3=α3−(β1,β1)(α3,β1)β1−(β2,β2)(α3,β2)β2 ,
…
β s = α s − ( α s , β 1 ) ( β 1 , β 1 ) β 1 − ( α s , β 2 ) ( β 2 , β 2 ) β 2 − ⋯ − ( α s , β s − 1 ) ( β s − 1 , β s − 1 ) β s − 1 \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} βs=αs−(β1,β1)(αs,β1)β1−(β2,β2)(αs,β2)β2−⋯−(βs−1,βs−1)(αs,βs−1)βs−1
(2)正交基和规范正交基
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。
2.4 线性方程组
2.4.1 克莱姆法则
线性方程组 { a 11 x 1 + a 12 x 2 + ⋯ + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ⋯ + a 2 n x n = b 2 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ a n 1 x 1 + a n 2 x 2 + ⋯ + a n n x n = b n \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} ⎩⎪⎪⎪⎨⎪⎪⎪⎧a11x1+a12x2+⋯+a1nxn=b1a21x1+a22x2+⋯+a2nxn=b2⋯⋯⋯⋯⋯⋯⋯⋯⋯an1x1+an2x2+⋯+annxn=bn
如果系数行列式 D = ∣ A ∣ ≠ 0 D = \left| A \right| \neq 0 D=∣A∣=0,则方程组有唯一解, x 1 = D 1 D , x 2 = D 2 D , ⋯ , x n = D n D x_{1} = \frac{D_{1}}{D},x_{2} = \frac{D_{2}}{D},\cdots,x_{n} =\frac{D_{n}}{D} x1=DD1,x2=DD2,⋯,xn=DDn,其中 D j D_{j} Dj是把 D D D中第 j j j列元素换成方程组右端的常数列所得的行列式。
2.4.2 方程组与矩阵关系
- n n n阶矩阵 A A A可逆 ⇔ A x = 0 \Leftrightarrow Ax = 0 ⇔Ax=0只有零解。 ⇔ ∀ b , A x = b \Leftrightarrow\forall b,Ax = b ⇔∀b,Ax=b总有唯一解,一般地, r ( A m × n ) = n ⇔ A x = 0 r(A_{m \times n}) = n \Leftrightarrow Ax= 0 r(Am×n)=n⇔Ax=0只有零解。
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
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设 A A A为 m × n m \times n m×n矩阵,若 r ( A m × n ) = m r(A_{m \times n}) = m r(Am×n)=m,则对 A x = b Ax =b Ax=b而言必有 r ( A ) = r ( A ⋮ b ) = m r(A) = r(A \vdots b) = m r(A)=r(A⋮b)=m,从而 A x = b Ax = b Ax=b有解。
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设 x 1 , x 2 , ⋯ x s x_{1},x_{2},\cdots x_{s} x1,x2,⋯xs为 A x = b Ax = b Ax=b的解,则 k 1 x 1 + k 2 x 2 ⋯ + k s x s k_{1}x_{1} + k_{2}x_{2}\cdots + k_{s}x_{s} k1x1+k2x2⋯+ksxs当 k 1 + k 2 + ⋯ + k s = 1 k_{1} + k_{2} + \cdots + k_{s} = 1 k1+k2+⋯+ks=1时仍为 A x = b Ax =b Ax=b的解;但当 k 1 + k 2 + ⋯ + k s = 0 k_{1} + k_{2} + \cdots + k_{s} = 0 k1+k2+⋯+ks=0时,则为 A x = 0 Ax =0 Ax=0的解。特别 x 1 + x 2 2 \frac{x_{1} + x_{2}}{2} 2x1+x2为 A x = b Ax = b Ax=b的解; 2 x 3 − ( x 1 + x 2 ) 2x_{3} - (x_{1} +x_{2}) 2x3−(x1+x2)为 A x = 0 Ax = 0 Ax=0的解。
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非齐次线性方程组 A x = b {Ax} = b Ax=b无解 ⇔ r ( A ) + 1 = r ( A ‾ ) ⇔ b \Leftrightarrow r(A) + 1 =r(\overline{A}) \Leftrightarrow b ⇔r(A)+1=r(A)⇔b不能由 A A A的列向量 α 1 , α 2 , ⋯ , α n \alpha_{1},\alpha_{2},\cdots,\alpha_{n} α1,α2,⋯,αn线性表示。
2.4.3 通解与解空间
齐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
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齐次方程组 A x = 0 {Ax} = 0 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 A x = 0 {Ax}= 0 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 ) n - r(A) n−r(A),解空间的一组基称为齐次方程组的基础解系。
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η 1 , η 2 , ⋯ , η t \eta_{1},\eta_{2},\cdots,\eta_{t} η1,η2,⋯,ηt是 A x = 0 {Ax} = 0 Ax=0的基础解系,即:
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η 1 , η 2 , ⋯ , η t \eta_{1},\eta_{2},\cdots,\eta_{t} η1,η2,⋯,ηt是 A x = 0 {Ax} = 0 Ax=0的解;
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η 1 , η 2 , ⋯ , η t \eta_{1},\eta_{2},\cdots,\eta_{t} η1,η2,⋯,ηt线性无关;
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A x = 0 {Ax} = 0 Ax=0的任一解都可以由 η 1 , η 2 , ⋯ , η t \eta_{1},\eta_{2},\cdots,\eta_{t} η1,η2,⋯,ηt线性表出.
k 1 η 1 + k 2 η 2 + ⋯ + k t η t k_{1}\eta_{1} + k_{2}\eta_{2} + \cdots + k_{t}\eta_{t} k1η1+k2η2+⋯+ktηt是 A x = 0 {Ax} = 0 Ax=0的通解,其中 k 1 , k 2 , ⋯ , k t k_{1},k_{2},\cdots,k_{t} k1,k2,⋯,kt是任意常数。
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2.5 特征值与特征向量
2.5.1 特征值与特征向量概念
-
设 λ \lambda λ是 A A A的一个特征值,则 k A , a A + b E , A 2 , A m , f ( A ) , A T , A − 1 , A ∗ {kA},{aA} + {bE},A^{2},A^{m},f(A),A^{T},A^{- 1},A^{*} kA,aA+bE,A2,Am,f(A),AT,A−1,A∗有一个特征值分别为
k λ , a λ + b , λ 2 , λ m , f ( λ ) , λ , λ − 1 , ∣ A ∣ λ , {kλ},{aλ} + b,\lambda^{2},\lambda^{m},f(\lambda),\lambda,\lambda^{- 1},\frac{|A|}{\lambda}, kλ,aλ+b,λ2,λm,f(λ),λ,λ−1,λ∣A∣,且对应特征向量相同( A T A^{T} AT 例外)。 -
若 λ 1 , λ 2 , ⋯ , λ n \lambda_{1},\lambda_{2},\cdots,\lambda_{n} λ1,λ2,⋯,λn为 A A A的 n n n个特征值,则 ∑ i = 1 n λ i = ∑ i = 1 n a i i , ∏ i = 1 n λ i = ∣ A ∣ \sum_{i= 1}^{n}\lambda_{i} = \sum_{i = 1}^{n}a_{{ii}},\prod_{i = 1}^{n}\lambda_{i}= |A| ∑i=1nλi=∑i=1naii,∏i=1nλi=∣A∣ ,从而 ∣ A ∣ ≠ 0 ⇔ A |A| \neq 0 \Leftrightarrow A ∣A∣=0⇔A没有特征值。
-
设 λ 1 , λ 2 , ⋯ , λ s \lambda_{1},\lambda_{2},\cdots,\lambda_{s} λ1,λ2,⋯,λs为 A A A的 s s s个特征值,对应特征向量为 α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs,若: α = k 1 α 1 + k 2 α 2 + ⋯ + k s α s \alpha = k_{1}\alpha_{1} + k_{2}\alpha_{2} + \cdots + k_{s}\alpha_{s} α=k1α1+k2α2+⋯+ksαs 。则: A n α = k 1 A n α 1 + k 2 A n α 2 + ⋯ + k s A n α s = k 1 λ 1 n α 1 + k 2 λ 2 n α 2 + ⋯ k s λ s n α s 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} Anα=k1Anα1+k2Anα2+⋯+ksAnαs=k1λ1nα1+k2λ2nα2+⋯ksλsnαs 。
2.5.2 相似变换
- 若
A
∼
B
A \sim B
A∼B,则
- A T ∼ B T , A − 1 ∼ B − 1 , , A ∗ ∼ B ∗ A^{T} \sim B^{T},A^{- 1} \sim B^{- 1},,A^{*} \sim B^{*} AT∼BT,A−1∼B−1,,A∗∼B∗
- ∣ A ∣ = ∣ B ∣ , ∑ i = 1 n A i i = ∑ i = 1 n b i i , r ( A ) = r ( B ) |A| = |B|,\sum_{i = 1}^{n}A_{{ii}} = \sum_{i =1}^{n}b_{{ii}},r(A) = r(B) ∣A∣=∣B∣,∑i=1nAii=∑i=1nbii,r(A)=r(B)
- ∣ λ E − A ∣ = ∣ λ E − B ∣ |\lambda E - A| = |\lambda E - B| ∣λE−A∣=∣λE−B∣,对 ∀ λ \forall\lambda ∀λ成立
2.5.3 相似对角化
矩阵可相似对角化的充分必要条件
- 设 A A A为 n n n阶方阵,则 A A A可对角化 ⇔ \Leftrightarrow ⇔对每个 k i k_{i} ki重根特征值 λ i \lambda_{i} λi,有 n − r ( λ i E − A ) = k i n-r(\lambda_{i}E - A) = k_{i} n−r(λiE−A)=ki
- 设 A A A可对角化,则由 P − 1 A P = Λ , P^{- 1}{AP} = \Lambda, P−1AP=Λ,有 A = P Λ P − 1 A = {PΛ}P^{-1} A=PΛP−1,从而 A n = P Λ n P − 1 A^{n} = P\Lambda^{n}P^{- 1} An=PΛnP−1
- 重要结论
- 若 A ∼ B , C ∼ D A \sim B,C \sim D A∼B,C∼D,则 [ A O O C ] ∼ [ B O O D ] \begin{bmatrix} A & O \\ O & C \\\end{bmatrix} \sim \begin{bmatrix} B & O \\ O & D \\\end{bmatrix} [AOOC]∼[BOOD].
- 若 A ∼ B A \sim B A∼B,则 f ( A ) ∼ f ( B ) , ∣ f ( A ) ∣ ∼ ∣ f ( B ) ∣ f(A) \sim f(B),\left| f(A) \right| \sim \left| f(B)\right| f(A)∼f(B),∣f(A)∣∼∣f(B)∣,其中 f ( A ) f(A) f(A)为关于 n n n阶方阵 A A A的多项式。
- 若 A A 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 A A)
2.5.4 实对称矩阵
实对称矩阵的特征值、特征向量及相似对角阵
(1)相似矩阵
设 A , B A,B A,B为两个 n n n阶方阵,如果存在一个可逆矩阵 P P P,使得 B = P − 1 A P B =P^{- 1}{AP} B=P−1AP成立,则称矩阵 A A A与 B B B相似,记为 A ∼ B A \sim B A∼B。
(2)性质
相似矩阵的性质:如果 A ∼ B A \sim B A∼B则有:
-
A T ∼ B T A^{T} \sim B^{T} AT∼BT
-
A − 1 ∼ B − 1 A^{- 1} \sim B^{- 1} A−1∼B−1 (若 A A A, B B B均可逆)
-
A k ∼ B k A^{k} \sim B^{k} Ak∼Bk ( k k k为正整数)
-
∣ λ E − A ∣ = ∣ λ E − B ∣ \left| {λE} - A \right| = \left| {λE} - B \right| ∣λE−A∣=∣λE−B∣,从而 A , B A,B A,B
有相同的特征值 -
∣ A ∣ = ∣ B ∣ \left| A \right| = \left| B \right| ∣A∣=∣B∣,从而 A , B A,B A,B同时可逆或者不可逆
-
秩 ( A ) = \left( A \right) = (A)=秩 ( B ) , ∣ λ E − A ∣ = ∣ λ E − B ∣ \left( B \right),\left| {λE} - A \right| =\left| {λE} - B \right| (B),∣λE−A∣=∣λE−B∣, A , B A,B A,B不一定相似
2.6 二次型
2.6.1 二次齐次函数
n \mathbf{n} n个变量 x 1 , x 2 , ⋯ , x n \mathbf{x}_{\mathbf{1}}\mathbf{,}\mathbf{x}_{\mathbf{2}}\mathbf{,\cdots,}\mathbf{x}_{\mathbf{n}} x1,x2,⋯,xn的二次齐次函数
f ( x 1 , x 2 , ⋯ , x n ) = ∑ i = 1 n ∑ j = 1 n a i j x i y j f(x_{1},x_{2},\cdots,x_{n}) = \sum_{i = 1}^{n}{\sum_{j =1}^{n}{a_{{ij}}x_{i}y_{j}}} f(x1,x2,⋯,xn)=∑i=1n∑j=1naijxiyj,其中 a i j = a j i ( i , j = 1 , 2 , ⋯ , n ) a_{{ij}} = a_{{ji}}(i,j =1,2,\cdots,n) aij=aji(i,j=1,2,⋯,n),称为 n n n元二次型,简称二次型. 若令
x = [ x 1 x 1 ⋮ x n ] , A = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋯ ⋯ ⋯ ⋯ a n 1 a n 2 ⋯ a n n ] 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} x= ⎣⎢⎢⎢⎡x1x1⋮xn⎦⎥⎥⎥⎤,A=⎣⎢⎢⎡a11a21⋯an1a12a22⋯an2⋯⋯⋯⋯a1na2n⋯ann⎦⎥⎥⎤
这二次型 f f f可改写成矩阵向量形式 f = x T A x f =x^{T}{Ax} f=xTAx。其中 A A A称为二次型矩阵,因为 a i j = a j i ( i , j = 1 , 2 , ⋯ , n ) a_{{ij}} =a_{{ji}}(i,j =1,2,\cdots,n) aij=aji(i,j=1,2,⋯,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 A A的秩称为二次型的秩。
2.6.2 相关形式
(1)惯性定理
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。
(2)标准形
二次型 f = ( x 1 , x 2 , ⋯ , x n ) = x T A x f = \left( x_{1},x_{2},\cdots,x_{n} \right) =x^{T}{Ax} f=(x1,x2,⋯,xn)=xTAx经过合同变换 x = C y x = {Cy} x=Cy化为 f = x T A x = y T C T A C f = x^{T}{Ax} =y^{T}C^{T}{AC} f=xTAx=yTCTAC
y = ∑ i = 1 r d i y i 2 y = \sum_{i = 1}^{r}{d_{i}y_{i}^{2}} y=∑i=1rdiyi2称为 f ( r ≤ n ) f(r \leq n) f(r≤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 ) r(A) r(A)唯一确定。
(3)规范形
任一实二次型 f f f都可经过合同变换化为规范形 f = z 1 2 + z 2 2 + ⋯ z p 2 − z p + 1 2 − ⋯ − z r 2 f = z_{1}^{2} + z_{2}^{2} + \cdots z_{p}^{2} - z_{p + 1}^{2} - \cdots -z_{r}^{2} f=z12+z22+⋯zp2−zp+12−⋯−zr2,其中 r r r为 A A A的秩, p p p为正惯性指数, r − p r -p r−p为负惯性指数,且规范型唯一。
2.6.3 转换为标准形
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
- 设 A A A正定 ⇒ k A ( k > 0 ) , A T , A − 1 , A ∗ \Rightarrow {kA}(k > 0),A^{T},A^{- 1},A^{*} ⇒kA(k>0),AT,A−1,A∗正定; ∣ A ∣ > 0 |A| >0 ∣A∣>0, A A A可逆; a i i > 0 a_{{ii}} > 0 aii>0,且 ∣ A i i ∣ > 0 |A_{{ii}}| > 0 ∣Aii∣>0
- A A A, B B B正定 ⇒ A + B \Rightarrow A +B ⇒A+B正定,但 A B {AB} AB, B A {BA} BA不一定正定
-
A
A
A正定
⇔
f
(
x
)
=
x
T
A
x
>
0
,
∀
x
≠
0
\Leftrightarrow f(x) = x^{T}{Ax} > 0,\forall x \neq 0
⇔f(x)=xTAx>0,∀x=0
⇔ A \Leftrightarrow A ⇔A的各阶顺序主子式全大于零
⇔ A \Leftrightarrow A ⇔A的所有特征值大于零
⇔ A \Leftrightarrow A ⇔A的正惯性指数为 n n n
⇔ \Leftrightarrow ⇔存在可逆阵 P P P使 A = P T P A = P^{T}P A=PTP
⇔ \Leftrightarrow ⇔存在正交矩阵 Q Q Q,使 Q T A Q = Q − 1 A Q = ( λ 1 ⋱ λ n ) , Q^{T}{AQ} = Q^{- 1}{AQ} =\begin{pmatrix} \lambda_{1} & & \\ \begin{matrix} & \\ & \\ \end{matrix} &\ddots & \\ & & \lambda_{n} \\ \end{pmatrix}, QTAQ=Q−1AQ=⎝⎜⎜⎛λ1⋱λn⎠⎟⎟⎞,
其中 λ i > 0 , i = 1 , 2 , ⋯ , n . \lambda_{i} > 0,i = 1,2,\cdots,n. λi>0,i=1,2,⋯,n.正定 ⇒ k A ( k > 0 ) , A T , A − 1 , A ∗ \Rightarrow {kA}(k >0),A^{T},A^{- 1},A^{*} ⇒kA(k>0),AT,A−1,A∗正定; ∣ A ∣ > 0 , A |A| > 0,A ∣A∣>0,A可逆; a i i > 0 a_{{ii}} >0 aii>0,且 ∣ A i i ∣ > 0 |A_{{ii}}| > 0 ∣Aii∣>0 。
§03 概率论和数理统计
3.1 随机事件和概率
3.1.1 事件关系与运算
(1) 子事件: A ⊂ B A \subset B A⊂B,若 A A A发生,则 B B B发生。
(2) 相等事件: A = B A = B A=B,即 A ⊂ B A \subset B A⊂B,且 B ⊂ A B \subset A B⊂A 。
(3) 和事件: A ⋃ B A\bigcup B A⋃B(或 A + B A + B A+B), A A A与 B B B中至少有一个发生。
(4) 差事件: A − B A - B A−B, A A A发生但 B B B不发生。
(5) 积事件: A ⋂ B A\bigcap B A⋂B(或 A B {AB} AB), A A A与 B B B同时发生。
(6) 互斥事件(互不相容): A ⋂ B A\bigcap B A⋂B= ∅ \varnothing ∅。
(7) 互逆事件(对立事件):
A
⋂
B
=
∅
,
A
⋃
B
=
Ω
,
A
=
B
ˉ
,
B
=
A
ˉ
A\bigcap B=\varnothing ,A\bigcup B=\Omega ,A=\bar{B},B=\bar{A}
A⋂B=∅,A⋃B=Ω,A=Bˉ,B=Aˉ
3.1.2 运算律
(1) 交换律:
A
⋃
B
=
B
⋃
A
,
A
⋂
B
=
B
⋂
A
A\bigcup B=B\bigcup A,A\bigcap B=B\bigcap A
A⋃B=B⋃A,A⋂B=B⋂A
(2) 结合律:
(
A
⋃
B
)
⋃
C
=
A
⋃
(
B
⋃
C
)
(A\bigcup B)\bigcup C=A\bigcup (B\bigcup C)
(A⋃B)⋃C=A⋃(B⋃C)
(3) 分配律:
(
A
⋂
B
)
⋂
C
=
A
⋂
(
B
⋂
C
)
(A\bigcap B)\bigcap C=A\bigcap (B\bigcap C)
(A⋂B)⋂C=A⋂(B⋂C)
3.1.3 德·摩根律
A ⋃ B ‾ = A ˉ ⋂ B ˉ \overline{A\bigcup B}=\bar{A}\bigcap \bar{B} A⋃B=Aˉ⋂Bˉ A ⋂ B ‾ = A ˉ ⋃ B ˉ \overline{A\bigcap B}=\bar{A}\bigcup \bar{B} A⋂B=Aˉ⋃Bˉ
3.1.4 完全事件组
A 1 A 2 ⋯ A n {{A}_{1}}{{A}_{2}}\cdots {{A}{n}} A1A2⋯An 两两互斥,且和事件为必然事件,即 A i ⋂ A j = ∅ , i ≠ j , ⋃ i = 1 n = Ω {A_i} \bigcap {A_j}=\varnothing, i \ne j ,\bigcup_{i=1}^{n} = \Omega Ai⋂Aj=∅,i=j,⋃i=1n=Ω
3.1.5 概率基本公式
(1)条件概率
P ( B ∣ A ) = P ( A B ) P ( A ) P(B|A)=\frac{P(AB)}{P(A)} P(B∣A)=P(A)P(AB),表示 A A A发生的条件下, B B B发生的概率。
(2)全概率公式
P ( A ) = ∑ i = 1 n P ( A ∣ B i ) P ( B i ) , B i B j = ∅ , i ≠ j , ⋃ n i = 1 B i = Ω 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 P(A)=i=1∑nP(A∣Bi)P(Bi),BiBj=∅,i=j,i=1⋃nBi=Ω
(3)Bayes公式
P
(
B
j
∣
A
)
=
P
(
A
∣
B
j
)
P
(
B
j
)
∑
i
=
1
n
P
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A
∣
B
i
)
P
(
B
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,
j
=
1
,
2
,
⋯
,
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
P(Bj∣A)=i=1∑nP(A∣Bi)P(Bi)P(A∣Bj)P(Bj),j=1,2,⋯,n
注:上述公式中事件
B
i
{{B}_{i}}
Bi的个数可为可列个。
(4)乘法公式
P
(
A
1
A
2
)
=
P
(
A
1
)
P
(
A
2
∣
A
1
)
=
P
(
A
2
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P
(
A
1
∣
A
2
)
P({{A}_{1}}{{A}_{2}})=P({{A}_{1}})P({{A}_{2}}|{{A}_{1}})=P({{A}_{2}})P({{A}_{1}}|{{A}_{2}})
P(A1A2)=P(A1)P(A2∣A1)=P(A2)P(A1∣A2)
P
(
A
1
A
2
⋯
A
n
)
=
P
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A
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)
P
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A
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∣
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)
P
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∣
A
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A
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)
⋯
P
(
A
n
∣
A
1
A
2
⋯
A
n
−
1
)
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}})
P(A1A2⋯An)=P(A1)P(A2∣A1)P(A3∣A1A2)⋯P(An∣A1A2⋯An−1)
3.1.6 事件的独立性
(1)独立性定义
(1) A A A与 B B B相互独立 ⇔ P ( A B ) = P ( A ) P ( B ) \Leftrightarrow P(AB)=P(A)P(B) ⇔P(AB)=P(A)P(B)
(2)
A
A
A,
B
B
B,
C
C
C两两独立
⇔
P
(
A
B
)
=
P
(
A
)
P
(
B
)
\Leftrightarrow P(AB)=P(A)P(B)
⇔P(AB)=P(A)P(B);
P
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B
C
)
=
P
(
B
)
P
(
C
)
P(BC)=P(B)P(C)
P(BC)=P(B)P(C) ;
P
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A
C
)
=
P
(
A
)
P
(
C
)
P(AC)=P(A)P(C)
P(AC)=P(A)P(C);
(3)
A
A
A,
B
B
B,
C
C
C相互独立
⇔
P
(
A
B
)
=
P
(
A
)
P
(
B
)
\Leftrightarrow P(AB)=P(A)P(B)
⇔P(AB)=P(A)P(B);
P
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B
C
)
=
P
(
B
)
P
(
C
)
P(BC)=P(B)P(C)
P(BC)=P(B)P(C) ;
P
(
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C
)
=
P
(
A
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P
(
C
)
P(AC)=P(A)P(C)
P(AC)=P(A)P(C) ;
P
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B
C
)
=
P
(
A
)
P
(
B
)
P
(
C
)
P(ABC)=P(A)P(B)P(C)
P(ABC)=P(A)P(B)P(C)
(2)独立重复试验
将某试验独立重复 n n n次,若每次实验中事件A发生的概率为 p p p,则 n n n次试验中 A A A发生 k k k次的概率为:
P ( X = k ) = C n k p k ( 1 − p ) n − k P(X=k)=C_{n}^{k}{{p}^{k}}{{(1-p)}^{n-k}} P(X=k)=Cnkpk(1−p)n−k
3.1.7 结论
( 1 ) P ( A ˉ ) = 1 − P ( A ) (1)P(\bar{A})=1-P(A) (1)P(Aˉ)=1−P(A)
(
2
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P
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⋃
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=
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+
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(2)P(A\bigcup B)=P(A)+P(B)-P(AB)
(2)P(A⋃B)=P(A)+P(B)−P(AB)
P
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P(A\bigcup B\bigcup C)=P(A)+P(B)+P(C)-P(AB)-P(BC)-P(AC)+P(ABC)
P(A⋃B⋃C)=P(A)+P(B)+P(C)−P(AB)−P(BC)−P(AC)+P(ABC)
( 3 ) P ( A − B ) = P ( A ) − P ( A B ) (3)P(A-B)=P(A)-P(AB) (3)P(A−B)=P(A)−P(AB)
(
4
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P
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=
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+
P
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,
(4)P(A\bar{B})=P(A)-P(AB),P(A)=P(AB)+P(A\bar{B}),
(4)P(ABˉ)=P(A)−P(AB),P(A)=P(AB)+P(ABˉ),
P
(
A
⋃
B
)
=
P
(
A
)
+
P
(
A
ˉ
B
)
=
P
(
A
B
)
+
P
(
A
B
ˉ
)
+
P
(
A
ˉ
B
)
P(A\bigcup B)=P(A)+P(\bar{A}B)=P(AB)+P(A\bar{B})+P(\bar{A}B)
P(A⋃B)=P(A)+P(AˉB)=P(AB)+P(ABˉ)+P(AˉB)
(5)条件概率 P ( ⋅ ∣ B ) P(\centerdot |B) P(⋅∣B)满足概率的所有性质,
例如:. P ( A ˉ 1 ∣ B ) = 1 − P ( A 1 ∣ B ) P({{\bar{A}}_{1}}|B)=1-P({{A}_{1}}|B) P(Aˉ1∣B)=1−P(A1∣B)
P ( A 1 ⋃ A 2 ∣ B ) = P ( A 1 ∣ B ) + P ( A 2 ∣ B ) − P ( A 1 A 2 ∣ B ) P({{A}_{1}}\bigcup {{A}_{2}}|B)=P({{A}_{1}}|B)+P({{A}_{2}}|B)-P({{A}_{1}}{{A}_{2}}|B) P(A1⋃A2∣B)=P(A1∣B)+P(A2∣B)−P(A1A2∣B)
P ( A 1 A 2 ∣ B ) = P ( A 1 ∣ B ) P ( A 2 ∣ A 1 B ) P({{A}_{1}}{{A}_{2}}|B)=P({{A}_{1}}|B)P({{A}_{2}}|{{A}_{1}}B) P(A1A2∣B)=P(A1∣B)P(A2∣A1B)
(6)若 A 1 , A 2 , ⋯ , A n {{A}_{1}},{{A}_{2}},\cdots ,{{A}_{n}} A1,A2,⋯,An相互独立,则 P ( ⋂ i = 1 n A i ) = ∏ i = 1 n P ( A i ) , P(\bigcap\limits_{i=1}^{n}{{{A}_{i}}})=\prod\limits_{i=1}^{n}{P({{A}_{i}})}, P(i=1⋂nAi)=i=1∏nP(Ai),
P ( ⋃ i = 1 n A i ) = ∏ i = 1 n ( 1 − P ( A i ) ) P(\bigcup\limits_{i=1}^{n}{{{A}_{i}}})=\prod\limits_{i=1}^{n}{(1-P({{A}_{i}}))} P(i=1⋃nAi)=i=1∏n(1−P(Ai))
(7)互斥、互逆与独立性之间的关系:
A
A
A与
B
B
B互逆
⇒
\Rightarrow
⇒
A
A
A与
B
B
B互斥,但反之不成立,
A
A
A与
B
B
B互斥(或互逆)且均非零概率事件
⇒
\Rightarrow
⇒
A
A
A 与
B
B
B 不独立.
(8)若 A 1 , A 2 , ⋯ , A m , B 1 , B 2 , ⋯ , B n {{A}_{1}},{{A}_{2}},\cdots ,{{A}_{m}},{{B}_{1}},{{B}_{2}},\cdots ,{{B}_{n}} A1,A2,⋯,Am,B1,B2,⋯,Bn 相互独立,则 f ( A 1 , A 2 , ⋯ , A m ) f({{A}_{1}},{{A}_{2}},\cdots ,{{A}_{m}}) f(A1,A2,⋯,Am) 与 g ( B 1 , B 2 , ⋯ , B n ) g({{B}_{1}},{{B}_{2}},\cdots ,{{B}_{n}}) g(B1,B2,⋯,Bn) 也相互独立,其中 f ( ⋅ ) , g ( ⋅ ) f(\centerdot ),g(\centerdot ) f(⋅),g(⋅) 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.
3.2 随机变量与概率分布
3.2.1 定义
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
3.2.2 概念与性质
-
定义: F ( x ) = P ( X ≤ x ) , − ∞ < x < + ∞ F(x) = P(X \leq x), - \infty < x < + \infty F(x)=P(X≤x),−∞<x<+∞
-
性质:
- 0 ≤ F ( x ) ≤ 1 0 \leq F(x) \leq 1 0≤F(x)≤1
- F ( x ) F(x) F(x)单调不减
- 右连续 F ( x + 0 ) = F ( x ) F(x + 0) = F(x) F(x+0)=F(x)
- F ( − ∞ ) = 0 , F ( + ∞ ) = 1 F( - \infty) = 0,F( + \infty) = 1 F(−∞)=0,F(+∞)=1
3.2.3 离散随机变量
P ( X = x i ) = p i , i = 1 , 2 , ⋯ , n , ⋯ p i ≥ 0 , ∑ i = 1 ∞ p i = 1 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 P(X=xi)=pi,i=1,2,⋯,n,⋯pi≥0,i=1∑∞pi=1
3.2.4 连续随机变量
概率密度 f ( x ) f(x) f(x);非负可积,且:
(1) f ( x ) ≥ 0 , f(x) \geq 0, f(x)≥0,
(2) ∫ − ∞ + ∞ f ( x ) d x = 1 \int_{- \infty}^{+\infty}{f(x){dx} = 1} ∫−∞+∞f(x)dx=1
(3) x x x为 f ( x ) f(x) f(x)的连续点,则:
f ( x ) = F ′ ( x ) f(x) = F'(x) f(x)=F′(x)分布函数 F ( x ) = ∫ − ∞ x f ( t ) d t F(x) = \int_{- \infty}^{x}{f(t){dt}} F(x)=∫−∞xf(t)dt
3.2.5 常见概率分布
(1) 0-1分布: P ( X = k ) = p k ( 1 − p ) 1 − k , k = 0 , 1 P(X = k) = p^{k}{(1 - p)}^{1 - k},k = 0,1 P(X=k)=pk(1−p)1−k,k=0,1
(2) 二项分布: B ( n , p ) B(n,p) B(n,p): P ( X = k ) = C n k p k ( 1 − p ) n − k , k = 0 , 1 , ⋯ , n P(X = k) = C_{n}^{k}p^{k}{(1 - p)}^{n - k},k =0,1,\cdots,n P(X=k)=Cnkpk(1−p)n−k,k=0,1,⋯,n
(3) Poisson分布: p ( λ ) p(\lambda) p(λ): P ( X = k ) = λ k k ! e − λ , λ > 0 , k = 0 , 1 , 2 ⋯ P(X = k) = \frac{\lambda^{k}}{k!}e^{-\lambda},\lambda > 0,k = 0,1,2\cdots P(X=k)=k!λke−λ,λ>0,k=0,1,2⋯
(4) 均匀分布 U ( a , b ) U(a,b) U(a,b): f ( x ) = { 1 b − a , a < x < b 0 , f(x) = \{ \begin{matrix} & \frac{1}{b - a},a < x< b \\ & 0, \\ \end{matrix} f(x)={b−a1,a<x<b0,
(5) 正态分布: N ( μ , σ 2 ) : N(\mu,\sigma^{2}): N(μ,σ2): φ ( x ) = 1 2 π σ e − ( x − μ ) 2 2 σ 2 , σ > 0 , ∞ < x < + ∞ \varphi(x) =\frac{1}{\sqrt{2\pi}\sigma}e^{- \frac{{(x - \mu)}^{2}}{2\sigma^{2}}},\sigma > 0,\infty < x < + \infty φ(x)=2πσ1e−2σ2(x−μ)2,σ>0,∞<x<+∞
(6)指数分布: E ( λ ) : f ( x ) = { λ e − λ x , x > 0 , λ > 0 0 , E(\lambda):f(x) =\{ \begin{matrix} & \lambda e^{-{λx}},x > 0,\lambda > 0 \\ & 0, \\ \end{matrix} E(λ):f(x)={λe−λx,x>0,λ>00,
(7)几何分布: G ( p ) : P ( X = k ) = ( 1 − p ) k − 1 p , 0 < p < 1 , k = 1 , 2 , ⋯ . G(p):P(X = k) = {(1 - p)}^{k - 1}p,0 < p < 1,k = 1,2,\cdots. G(p):P(X=k)=(1−p)k−1p,0<p<1,k=1,2,⋯.
(8)超几何分布: H ( N , M , n ) : P ( X = k ) = C M k C N − M n − k C N n , k = 0 , 1 , ⋯ , m i n ( n , M ) 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) H(N,M,n):P(X=k)=CNnCMkCN−Mn−k,k=0,1,⋯,min(n,M)
3.2.6 随机变量函数概率分布
(1)离散型
P ( X = x 1 ) = p i , Y = g ( X ) P(X = x_{1}) = p_{i},Y = g(X) P(X=x1)=pi,Y=g(X)
则: P ( Y = y j ) = ∑ g ( x i ) = y i P ( X = x i ) P(Y = y_{j}) = \sum_{g(x_{i}) = y_{i}}^{}{P(X = x_{i})} P(Y=yj)=∑g(xi)=yiP(X=xi)
(2)连续型
X ~ f X ( x ) , Y = g ( x ) X\tilde{\ }f_{X}(x),Y = g(x) X ~fX(x),Y=g(x)
则: F y ( y ) = P ( Y ≤ y ) = P ( g ( X ) ≤ y ) = ∫ g ( x ) ≤ y f x ( x ) d x F_{y}(y) = P(Y \leq y) = P(g(X) \leq y) = \int_{g(x) \leq y}^{}{f_{x}(x)dx} Fy(y)=P(Y≤y)=P(g(X)≤y)=∫g(x)≤yfx(x)dx, f Y ( y ) = F Y ′ ( y ) f_{Y}(y) = F'_{Y}(y) fY(y)=FY′(y)
3.2.7 重要公式与结论
(1) X ∼ N ( 0 , 1 ) ⇒ φ ( 0 ) = 1 2 π , Φ ( 0 ) = 1 2 , X\sim N(0,1) \Rightarrow \varphi(0) = \frac{1}{\sqrt{2\pi}},\Phi(0) =\frac{1}{2}, X∼N(0,1)⇒φ(0)=2π1,Φ(0)=21, Φ ( − a ) = P ( X ≤ − a ) = 1 − Φ ( a ) \Phi( - a) = P(X \leq - a) = 1 - \Phi(a) Φ(−a)=P(X≤−a)=1−Φ(a)
(2) X ∼ N ( μ , σ 2 ) ⇒ X − μ σ ∼ N ( 0 , 1 ) , P ( X ≤ a ) = Φ ( a − μ σ ) 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}) X∼N(μ,σ2)⇒σX−μ∼N(0,1),P(X≤a)=Φ(σa−μ)
(3) X ∼ E ( λ ) ⇒ P ( X > s + t ∣ X > s ) = P ( X > t ) X\sim E(\lambda) \Rightarrow P(X > s + t|X > s) = P(X > t) X∼E(λ)⇒P(X>s+t∣X>s)=P(X>t)
(4) X ∼ G ( p ) ⇒ P ( X = m + k ∣ X > m ) = P ( X = k ) X\sim G(p) \Rightarrow P(X = m + k|X > m) = P(X = k) X∼G(p)⇒P(X=m+k∣X>m)=P(X=k)
(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。
(6) 存在既非离散也非连续型随机变量。
3.2.8 多维随机变量
1.二维随机变量及其联合分布
由两个随机变量构成的随机向量 ( X , Y ) (X,Y) (X,Y), 联合分布为 F ( x , y ) = P ( X ≤ x , Y ≤ y ) F(x,y) = P(X \leq x,Y \leq y) F(x,y)=P(X≤x,Y≤y)
2.二维离散型随机变量的分布
(1) 联合概率分布律 P { X = x i , Y = y j } = p i j ; i , j = 1 , 2 , ⋯ P\{ X = x_{i},Y = y_{j}\} = p_{{ij}};i,j =1,2,\cdots P{X=xi,Y=yj}=pij;i,j=1,2,⋯
(2) 边缘分布律 p i ⋅ = ∑ j = 1 ∞ p i j , i = 1 , 2 , ⋯ p_{i \cdot} = \sum_{j = 1}^{\infty}p_{{ij}},i =1,2,\cdots pi⋅=∑j=1∞pij,i=1,2,⋯ p ⋅ j = ∑ i ∞ p i j , j = 1 , 2 , ⋯ p_{\cdot j} = \sum_{i}^{\infty}p_{{ij}},j = 1,2,\cdots p⋅j=∑i∞pij,j=1,2,⋯
(3) 条件分布律
P
{
X
=
x
i
∣
Y
=
y
j
}
=
p
i
j
p
⋅
j
P\{ X = x_{i}|Y = y_{j}\} = \frac{p_{{ij}}}{p_{\cdot j}}
P{X=xi∣Y=yj}=p⋅jpij
P
{
Y
=
y
j
∣
X
=
x
i
}
=
p
i
j
p
i
⋅
P\{ Y = y_{j}|X = x_{i}\} = \frac{p_{{ij}}}{p_{i \cdot}}
P{Y=yj∣X=xi}=pi⋅pij
3. 二维连续性随机变量的密度
(1) 联合概率密度 f ( x , y ) : f(x,y): f(x,y):
-
f ( x , y ) ≥ 0 f(x,y) \geq 0 f(x,y)≥0
-
∫ − ∞ + ∞ ∫ − ∞ + ∞ f ( x , y ) d x d y = 1 \int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{f(x,y)dxdy}} = 1 ∫−∞+∞∫−∞+∞f(x,y)dxdy=1
(2) 分布函数: F ( x , y ) = ∫ − ∞ x ∫ − ∞ y f ( u , v ) d u d v F(x,y) = \int_{- \infty}^{x}{\int_{- \infty}^{y}{f(u,v)dudv}} F(x,y)=∫−∞x∫−∞yf(u,v)dudv
(3) 边缘概率密度: f X ( x ) = ∫ − ∞ + ∞ f ( x , y ) d y f_{X}\left( x \right) = \int_{- \infty}^{+ \infty}{f\left( x,y \right){dy}} fX(x)=∫−∞+∞f(x,y)dy f Y ( y ) = ∫ − ∞ + ∞ f ( x , y ) d x f_{Y}(y) = \int_{- \infty}^{+ \infty}{f(x,y)dx} fY(y)=∫−∞+∞f(x,y)dx
(4) 条件概率密度: f X ∣ Y ( x | y ) = f ( x , y ) f Y ( y ) f_{X|Y}\left( x \middle| y \right) = \frac{f\left( x,y \right)}{f_{Y}\left( y \right)} fX∣Y(x∣y)=fY(y)f(x,y) f Y ∣ X ( y ∣ x ) = f ( x , y ) f X ( x ) f_{Y|X}(y|x) = \frac{f(x,y)}{f_{X}(x)} fY∣X(y∣x)=fX(x)f(x,y)
4.常见二维随机变量的联合分布
(1) 二维均匀分布: ( x , y ) ∼ U ( D ) (x,y) \sim U(D) (x,y)∼U(D) , f ( x , y ) = { 1 S ( D ) , ( x , y ) ∈ D 0 , 其 他 f(x,y) = \begin{cases} \frac{1}{S(D)},(x,y) \in D \\ 0,其他 \end{cases} f(x,y)={S(D)1,(x,y)∈D0,其他
(2) 二维正态分布: ( X , Y ) ∼ N ( μ 1 , μ 2 , σ 1 2 , σ 2 2 , ρ ) (X,Y)\sim N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho) (X,Y)∼N(μ1,μ2,σ12,σ22,ρ), ( X , Y ) ∼ N ( μ 1 , μ 2 , σ 1 2 , σ 2 2 , ρ ) (X,Y)\sim N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho) (X,Y)∼N(μ1,μ2,σ12,σ22,ρ)
f ( x , y ) = 1 2 π σ 1 σ 2 1 − ρ 2 . exp { − 1 2 ( 1 − ρ 2 ) [ ( x − μ 1 ) 2 σ 1 2 − 2 ρ ( x − μ 1 ) ( y − μ 2 ) σ 1 σ 2 + ( y − μ 2 ) 2 σ 2 2 ] } 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\} f(x,y)=2πσ1σ21−ρ21.exp{2(1−ρ2)−1[σ12(x−μ1)2−2ρσ1σ2(x−μ1)(y−μ2)+σ22(y−μ2)2]}
5.随机变量的独立性和相关性
X X X和 Y Y Y的相互独立: ⇔ F ( x , y ) = F X ( x ) F Y ( y ) \Leftrightarrow F\left( x,y \right) = F_{X}\left( x \right)F_{Y}\left( y \right) ⇔F(x,y)=FX(x)FY(y):
⇔
p
i
j
=
p
i
⋅
⋅
p
⋅
j
\Leftrightarrow p_{{ij}} = p_{i \cdot} \cdot p_{\cdot j}
⇔pij=pi⋅⋅p⋅j(离散型)
⇔
f
(
x
,
y
)
=
f
X
(
x
)
f
Y
(
y
)
\Leftrightarrow f\left( x,y \right) = f_{X}\left( x \right)f_{Y}\left( y \right)
⇔f(x,y)=fX(x)fY(y)(连续型)
X X X和 Y Y Y的相关性:
相关系数
ρ
X
Y
=
0
\rho_{{XY}} = 0
ρXY=0时,称
X
X
X和
Y
Y
Y不相关,
否则称
X
X
X和
Y
Y
Y相关
6.两个随机变量简单函数的概率分布
离散型: P ( X = x i , Y = y i ) = p i j , Z = g ( X , Y ) P\left( X = x_{i},Y = y_{i} \right) = p_{{ij}},Z = g\left( X,Y \right) P(X=xi,Y=yi)=pij,Z=g(X,Y) 则:
P ( Z = z k ) = P { g ( X , Y ) = z k } = ∑ g ( x i , y i ) = z k P ( X = x i , Y = y j ) 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)} P(Z=zk)=P{g(X,Y)=zk}=∑g(xi,yi)=zkP(X=xi,Y=yj)
连续型:
(
X
,
Y
)
∼
f
(
x
,
y
)
,
Z
=
g
(
X
,
Y
)
\left( X,Y \right) \sim f\left( x,y \right),Z = g\left( X,Y \right)
(X,Y)∼f(x,y),Z=g(X,Y)
则:
F z ( z ) = P { g ( X , Y ) ≤ z } = ∬ g ( x , y ) ≤ z f ( x , y ) d x d y 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} Fz(z)=P{g(X,Y)≤z}=∬g(x,y)≤zf(x,y)dxdy, f z ( z ) = F z ′ ( z ) f_{z}(z) = F'_{z}(z) fz(z)=Fz′(z)
7.重要公式与结论
(1) 边缘密度公式:
f
X
(
x
)
=
∫
−
∞
+
∞
f
(
x
,
y
)
d
y
,
f_{X}(x) = \int_{- \infty}^{+ \infty}{f(x,y)dy,}
fX(x)=∫−∞+∞f(x,y)dy,
f
Y
(
y
)
=
∫
−
∞
+
∞
f
(
x
,
y
)
d
x
f_{Y}(y) = \int_{- \infty}^{+ \infty}{f(x,y)dx}
fY(y)=∫−∞+∞f(x,y)dx
(2) P { ( X , Y ) ∈ D } = ∬ D f ( x , y ) d x d y P\left\{ \left( X,Y \right) \in D \right\} = \iint_{D}^{}{f\left( x,y \right){dxdy}} P{(X,Y)∈D}=∬Df(x,y)dxdy
(3) 若
(
X
,
Y
)
(X,Y)
(X,Y)服从二维正态分布
N
(
μ
1
,
μ
2
,
σ
1
2
,
σ
2
2
,
ρ
)
N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho)
N(μ1,μ2,σ12,σ22,ρ)
则有:
-
X ∼ N ( μ 1 , σ 1 2 ) , Y ∼ N ( μ 2 , σ 2 2 ) . X\sim N\left( \mu_{1},\sigma_{1}^{2} \right),Y\sim N(\mu_{2},\sigma_{2}^{2}). X∼N(μ1,σ12),Y∼N(μ2,σ22).
-
X X X与 Y Y Y相互独立 ⇔ ρ = 0 \Leftrightarrow \rho = 0 ⇔ρ=0,即 X X X与 Y Y Y不相关。
-
C 1 X + C 2 Y ∼ N ( C 1 μ 1 + C 2 μ 2 , C 1 2 σ 1 2 + C 2 2 σ 2 2 + 2 C 1 C 2 σ 1 σ 2 ρ ) 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) C1X+C2Y∼N(C1μ1+C2μ2,C12σ12+C22σ22+2C1C2σ1σ2ρ)
-
X {\ X} X关于 Y = y Y=y Y=y的条件分布为: N ( μ 1 + ρ σ 1 σ 2 ( y − μ 2 ) , σ 1 2 ( 1 − ρ 2 ) ) N(\mu_{1} + \rho\frac{\sigma_{1}}{\sigma_{2}}(y - \mu_{2}),\sigma_{1}^{2}(1 - \rho^{2})) N(μ1+ρσ2σ1(y−μ2),σ12(1−ρ2))
-
Y Y Y关于 X = x X = x X=x的条件分布为: N ( μ 2 + ρ σ 2 σ 1 ( x − μ 1 ) , σ 2 2 ( 1 − ρ 2 ) ) N(\mu_{2} + \rho\frac{\sigma_{2}}{\sigma_{1}}(x - \mu_{1}),\sigma_{2}^{2}(1 - \rho^{2})) N(μ2+ρσ1σ2(x−μ1),σ22(1−ρ2))
(4) 若
X
X
X与
Y
Y
Y独立,且分别服从
N
(
μ
1
,
σ
1
2
)
,
N
(
μ
1
,
σ
2
2
)
,
N(\mu_{1},\sigma_{1}^{2}),N(\mu_{1},\sigma_{2}^{2}),
N(μ1,σ12),N(μ1,σ22),
则:
(
X
,
Y
)
∼
N
(
μ
1
,
μ
2
,
σ
1
2
,
σ
2
2
,
0
)
,
\left( X,Y \right)\sim N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},0),
(X,Y)∼N(μ1,μ2,σ12,σ22,0),
C 1 X + C 2 Y ~ N ( C 1 μ 1 + C 2 μ 2 , C 1 2 σ 1 2 C 2 2 σ 2 2 ) . 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}). C1X+C2Y ~N(C1μ1+C2μ2,C12σ12C22σ22).
(5) 若 X X X与 Y Y Y相互独立, f ( x ) f\left( x \right) f(x)和 g ( x ) g\left( x \right) g(x)为连续函数, 则 f ( X ) f\left( X \right) f(X)和 g ( Y ) g(Y) g(Y)也相互独立。
3.2.9 随机变量数字特征
1.数学期望
离散型: P { X = x i } = p i , E ( X ) = ∑ i x i p i P\left\{ X = x_{i} \right\} = p_{i},E(X) = \sum_{i}^{}{x_{i}p_{i}} P{X=xi}=pi,E(X)=∑ixipi;
连续型: X ∼ f ( x ) , E ( X ) = ∫ − ∞ + ∞ x f ( x ) d x X\sim f(x),E(X) = \int_{- \infty}^{+ \infty}{xf(x)dx} X∼f(x),E(X)=∫−∞+∞xf(x)dx
性质:
(1) E ( C ) = C , E [ E ( X ) ] = E ( X ) E(C) = C,E\lbrack E(X)\rbrack = E(X) E(C)=C,E[E(X)]=E(X)
(2) E ( C 1 X + C 2 Y ) = C 1 E ( X ) + C 2 E ( Y ) E(C_{1}X + C_{2}Y) = C_{1}E(X) + C_{2}E(Y) E(C1X+C2Y)=C1E(X)+C2E(Y)
(3) 若 X X X和 Y Y Y独立,则 E ( X Y ) = E ( X ) E ( Y ) E(XY) = E(X)E(Y) E(XY)=E(X)E(Y)
(4) [ E ( X Y ) ] 2 ≤ E ( X 2 ) E ( Y 2 ) \left\lbrack E(XY) \right\rbrack^{2} \leq E(X^{2})E(Y^{2}) [E(XY)]2≤E(X2)E(Y2)
2.方差: D ( X ) = E [ X − E ( X ) ] 2 = E ( X 2 ) − [ E ( X ) ] 2 D(X) = E\left\lbrack X - E(X) \right\rbrack^{2} = E(X^{2}) - \left\lbrack E(X) \right\rbrack^{2} D(X)=E[X−E(X)]2=E(X2)−[E(X)]2
3.标准差: D ( X ) \sqrt{D(X)} D(X),
4.离散型: D ( X ) = ∑ i [ x i − E ( X ) ] 2 p i D(X) = \sum_{i}^{}{\left\lbrack x_{i} - E(X) \right\rbrack^{2}p_{i}} D(X)=∑i[xi−E(X)]2pi
5.连续型: D ( X ) = ∫ − ∞ + ∞ [ x − E ( X ) ] 2 f ( x ) d x D(X) = {\int_{- \infty}^{+ \infty}\left\lbrack x - E(X) \right\rbrack}^{2}f(x)dx D(X)=∫−∞+∞[x−E(X)]2f(x)dx
性质:
(1) D ( C ) = 0 , D [ E ( X ) ] = 0 , D [ D ( X ) ] = 0 \ D(C) = 0,D\lbrack E(X)\rbrack = 0,D\lbrack D(X)\rbrack = 0 D(C)=0,D[E(X)]=0,D[D(X)]=0
(2) X X X与 Y Y Y相互独立,则 D ( X ± Y ) = D ( X ) + D ( Y ) D(X \pm Y) = D(X) + D(Y) D(X±Y)=D(X)+D(Y)
(3) D ( C 1 X + C 2 ) = C 1 2 D ( X ) \ D\left( C_{1}X + C_{2} \right) = C_{1}^{2}D\left( X \right) D(C1X+C2)=C12D(X)
(4) 一般有 D ( X ± Y ) = D ( X ) + D ( Y ) ± 2 C o v ( X , Y ) = D ( X ) + D ( Y ) ± 2 ρ D ( X ) D ( Y ) 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)} D(X±Y)=D(X)+D(Y)±2Cov(X,Y)=D(X)+D(Y)±2ρD(X)D(Y)
(5) D ( X ) < E ( X − C ) 2 , C ≠ E ( X ) \ D\left( X \right) < E\left( X - C \right)^{2},C \neq E\left( X \right) D(X)<E(X−C)2,C=E(X)
(6) D ( X ) = 0 ⇔ P { X = C } = 1 \ D(X) = 0 \Leftrightarrow P\left\{ X = C \right\} = 1 D(X)=0⇔P{X=C}=1
6.随机变量函数的数学期望
(1) 对于函数 Y = g ( x ) Y = g(x) Y=g(x)
X X X为离散型: P { X = x i } = p i , E ( Y ) = ∑ i g ( x i ) p i P\{ X = x_{i}\} = p_{i},E(Y) = \sum_{i}^{}{g(x_{i})p_{i}} P{X=xi}=pi,E(Y)=∑ig(xi)pi;
X X X为连续型: X ∼ f ( x ) , E ( Y ) = ∫ − ∞ + ∞ g ( x ) f ( x ) d x X\sim f(x),E(Y) = \int_{- \infty}^{+ \infty}{g(x)f(x)dx} X∼f(x),E(Y)=∫−∞+∞g(x)f(x)dx
(2) Z = g ( X , Y ) Z = g(X,Y) Z=g(X,Y); ( X , Y ) ∼ P { X = x i , Y = y j } = p i j \left( X,Y \right)\sim P\{ X = x_{i},Y = y_{j}\} = p_{{ij}} (X,Y)∼P{X=xi,Y=yj}=pij; E ( Z ) = ∑ i ∑ j g ( x i , y j ) p i j E(Z) = \sum_{i}^{}{\sum_{j}^{}{g(x_{i},y_{j})p_{{ij}}}} E(Z)=∑i∑jg(xi,yj)pij ( X , Y ) ∼ f ( x , y ) \left( X,Y \right)\sim f(x,y) (X,Y)∼f(x,y); E ( Z ) = ∫ − ∞ + ∞ ∫ − ∞ + ∞ g ( x , y ) f ( x , y ) d x d y E(Z) = \int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{g(x,y)f(x,y)dxdy}} E(Z)=∫−∞+∞∫−∞+∞g(x,y)f(x,y)dxdy
7.协方差
C o v ( X , Y ) = E [ ( X − E ( X ) ( Y − E ( Y ) ) ] Cov(X,Y) = E\left\lbrack (X - E(X)(Y - E(Y)) \right\rbrack Cov(X,Y)=E[(X−E(X)(Y−E(Y))]
8.相关系数
ρ
X
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C
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(
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\rho_{{XY}} = \frac{Cov(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}}
ρXY=D(X)D(Y)Cov(X,Y),
k
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k阶原点矩
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E(X^{k})
E(Xk);
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k阶中心矩
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E
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k
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E\left\{ {\lbrack X - E(X)\rbrack}^{k} \right\}
E{[X−E(X)]k}
性质:
(1) C o v ( X , Y ) = C o v ( Y , X ) \ Cov(X,Y) = Cov(Y,X) Cov(X,Y)=Cov(Y,X)
(2) C o v ( a X , b Y ) = a b C o v ( Y , X ) \ Cov(aX,bY) = abCov(Y,X) Cov(aX,bY)=abCov(Y,X)
(3) C o v ( X 1 + X 2 , Y ) = C o v ( X 1 , Y ) + C o v ( X 2 , Y ) \ Cov(X_{1} + X_{2},Y) = Cov(X_{1},Y) + Cov(X_{2},Y) Cov(X1+X2,Y)=Cov(X1,Y)+Cov(X2,Y)
(4) ∣ ρ ( X , Y ) ∣ ≤ 1 \ \left| \rho\left( X,Y \right) \right| \leq 1 ∣ρ(X,Y)∣≤1
(5) ρ ( X , Y ) = 1 ⇔ P ( Y = a X + b ) = 1 \ \rho\left( X,Y \right) = 1 \Leftrightarrow P\left( Y = aX + b \right) = 1 ρ(X,Y)=1⇔P(Y=aX+b)=1 ,其中 a > 0 a > 0 a>0
ρ
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X
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Y
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=
−
1
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\rho\left( X,Y \right) = - 1 \Leftrightarrow P\left( Y = aX + b \right) = 1
ρ(X,Y)=−1⇔P(Y=aX+b)=1
,其中
a
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0
a < 0
a<0
9.重要公式与结论
(1) D ( X ) = E ( X 2 ) − E 2 ( X ) \ D(X) = E(X^{2}) - E^{2}(X) D(X)=E(X2)−E2(X)
(2) C o v ( X , Y ) = E ( X Y ) − E ( X ) E ( Y ) \ Cov(X,Y) = E(XY) - E(X)E(Y) Cov(X,Y)=E(XY)−E(X)E(Y)
(3) ∣ ρ ( X , Y ) ∣ ≤ 1 , \left| \rho\left( X,Y \right) \right| \leq 1, ∣ρ(X,Y)∣≤1,且 ρ ( X , Y ) = 1 ⇔ P ( Y = a X + b ) = 1 \rho\left( X,Y \right) = 1 \Leftrightarrow P\left( Y = aX + b \right) = 1 ρ(X,Y)=1⇔P(Y=aX+b)=1,其中 a > 0 a > 0 a>0
ρ ( X , Y ) = − 1 ⇔ P ( Y = a X + b ) = 1 \rho\left( X,Y \right) = - 1 \Leftrightarrow P\left( Y = aX + b \right) = 1 ρ(X,Y)=−1⇔P(Y=aX+b)=1,其中 a < 0 a < 0 a<0
(4) 下面5个条件互为充要条件:
ρ ( X , Y ) = 0 \rho(X,Y) = 0 ρ(X,Y)=0 ⇔ C o v ( X , Y ) = 0 \Leftrightarrow Cov(X,Y) = 0 ⇔Cov(X,Y)=0 ⇔ E ( X , Y ) = E ( X ) E ( Y ) \Leftrightarrow E(X,Y) = E(X)E(Y) ⇔E(X,Y)=E(X)E(Y) ⇔ D ( X + Y ) = D ( X ) + D ( Y ) \Leftrightarrow D(X + Y) = D(X) + D(Y) ⇔D(X+Y)=D(X)+D(Y) ⇔ D ( X − Y ) = D ( X ) + D ( Y ) \Leftrightarrow D(X - Y) = D(X) + D(Y) ⇔D(X−Y)=D(X)+D(Y)
注: X X X与 Y Y 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。
3.3 数理统计
总体:研究对象的全体,它是一个随机变量,用 X X X表示。
个体:组成总体的每个基本元素。
简单随机样本:来自总体 X X X的 n n n个相互独立且与总体同分布的随机变量 X 1 , X 2 ⋯ , X n X_{1},X_{2}\cdots,X_{n} X1,X2⋯,Xn,称为容量为 n n n的简单随机样本,简称样本。
统计量:设 X 1 , X 2 ⋯ , X n , X_{1},X_{2}\cdots,X_{n}, X1,X2⋯,Xn,是来自总体 X X X的一个样本, g ( X 1 , X 2 ⋯ , X n ) g(X_{1},X_{2}\cdots,X_{n}) g(X1,X2⋯,Xn))是样本的连续函数,且 g ( ) g() g()中不含任何未知参数,则称 g ( X 1 , X 2 ⋯ , X n ) g(X_{1},X_{2}\cdots,X_{n}) g(X1,X2⋯,Xn)为统计量。
样本均值: X ‾ = 1 n ∑ i = 1 n X i \overline{X} = \frac{1}{n}\sum_{i = 1}^{n}X_{i} X=n1∑i=1nXi
样本方差: S 2 = 1 n − 1 ∑ i = 1 n ( X i − X ‾ ) 2 S^{2} = \frac{1}{n - 1}\sum_{i = 1}^{n}{(X_{i} - \overline{X})}^{2} S2=n−11∑i=1n(Xi−X)2
样本矩:样本 k k k阶原点矩: A k = 1 n ∑ i = 1 n X i k , k = 1 , 2 , ⋯ A_{k} = \frac{1}{n}\sum_{i = 1}^{n}X_{i}^{k},k = 1,2,\cdots Ak=n1∑i=1nXik,k=1,2,⋯
样本 k k k阶中心矩: B k = 1 n ∑ i = 1 n ( X i − X ‾ ) k , k = 1 , 2 , ⋯ B_{k} = \frac{1}{n}\sum_{i = 1}^{n}{(X_{i} - \overline{X})}^{k},k = 1,2,\cdots Bk=n1∑i=1n(Xi−X)k,k=1,2,⋯
2.分布
χ 2 \chi^{2} χ2分布: χ 2 = X 1 2 + X 2 2 + ⋯ + X n 2 ∼ χ 2 ( n ) \chi^{2} = X_{1}^{2} + X_{2}^{2} + \cdots + X_{n}^{2}\sim\chi^{2}(n) χ2=X12+X22+⋯+Xn2∼χ2(n),其中 X 1 , X 2 ⋯ , X n , X_{1},X_{2}\cdots,X_{n}, X1,X2⋯,Xn,相互独立,且同服从 N ( 0 , 1 ) N(0,1) N(0,1)
t t t分布: T = X Y / n ∼ t ( n ) T = \frac{X}{\sqrt{Y/n}}\sim t(n) T=Y/nX∼t(n) ,其中 X ∼ N ( 0 , 1 ) , Y ∼ χ 2 ( n ) , X\sim N\left( 0,1 \right),Y\sim\chi^{2}(n), X∼N(0,1),Y∼χ2(n),且 X X X, Y Y Y 相互独立。
F F F分布: F = X / n 1 Y / n 2 ∼ F ( n 1 , n 2 ) F = \frac{X/n_{1}}{Y/n_{2}}\sim F(n_{1},n_{2}) F=Y/n2X/n1∼F(n1,n2),其中 X ∼ χ 2 ( n 1 ) , Y ∼ χ 2 ( n 2 ) , X\sim\chi^{2}\left( n_{1} \right),Y\sim\chi^{2}(n_{2}), X∼χ2(n1),Y∼χ2(n2),且 X X X, Y Y Y相互独立。
分位数:若 P ( X ≤ x α ) = α , P(X \leq x_{\alpha}) = \alpha, P(X≤xα)=α,则称 x α x_{\alpha} xα为 X X X的 α \alpha α分位数
3.正态总体的常用样本分布
(1) 设 X 1 , X 2 ⋯ , X n X_{1},X_{2}\cdots,X_{n} X1,X2⋯,Xn为来自正态总体 N ( μ , σ 2 ) N(\mu,\sigma^{2}) N(μ,σ2)的样本,
X ‾ = 1 n ∑ i = 1 n X i , S 2 = 1 n − 1 ∑ i = 1 n ( X i − X ‾ ) 2 , \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},} X=n1∑i=1nXi,S2=n−11∑i=1n(Xi−X)2,则:
-
X ‾ ∼ N ( μ , σ 2 n ) \overline{X}\sim N\left( \mu,\frac{\sigma^{2}}{n} \right){\ \ } X∼N(μ,nσ2) 或者 X ‾ − μ σ n ∼ N ( 0 , 1 ) \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}}\sim N(0,1) nσX−μ∼N(0,1)
-
( n − 1 ) S 2 σ 2 = 1 σ 2 ∑ i = 1 n ( X i − X ‾ ) 2 ∼ χ 2 ( n − 1 ) \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)} σ2(n−1)S2=σ21∑i=1n(Xi−X)2∼χ2(n−1)
-
1 σ 2 ∑ i = 1 n ( X i − μ ) 2 ∼ χ 2 ( n ) \frac{1}{\sigma^{2}}\sum_{i = 1}^{n}{{(X_{i} - \mu)}^{2}\sim\chi^{2}(n)} σ21∑i=1n(Xi−μ)2∼χ2(n)
4) X ‾ − μ S / n ∼ t ( n − 1 ) {\ \ }\frac{\overline{X} - \mu}{S/\sqrt{n}}\sim t(n - 1) S/nX−μ∼t(n−1)
4.重要公式与结论
(1) 对于 χ 2 ∼ χ 2 ( n ) \chi^{2}\sim\chi^{2}(n) χ2∼χ2(n),有 E ( χ 2 ( n ) ) = n , D ( χ 2 ( n ) ) = 2 n ; E(\chi^{2}(n)) = n,D(\chi^{2}(n)) = 2n; E(χ2(n))=n,D(χ2(n))=2n;
(2) 对于 T ∼ t ( n ) T\sim t(n) T∼t(n),有 E ( T ) = 0 , D ( T ) = n n − 2 ( n > 2 ) E(T) = 0,D(T) = \frac{n}{n - 2}(n > 2) E(T)=0,D(T)=n−2n(n>2);
(3) 对于 F ~ F ( m , n ) F\tilde{\ }F(m,n) F ~F(m,n),有 1 F ∼ F ( n , m ) , F a / 2 ( m , n ) = 1 F 1 − a / 2 ( n , m ) ; \frac{1}{F}\sim F(n,m),F_{a/2}(m,n) = \frac{1}{F_{1 - a/2}(n,m)}; F1∼F(n,m),Fa/2(m,n)=F1−a/2(n,m)1;
(4) 对于任意总体 X X X,有 E ( X ‾ ) = E ( X ) , E ( S 2 ) = D ( X ) , D ( X ‾ ) = D ( X ) n E(\overline{X}) = E(X),E(S^{2}) = D(X),D(\overline{X}) = \frac{D(X)}{n} E(X)=E(X),E(S2)=D(X),D(X)=nD(X)
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