Math

机器学习的数学基础

高等数学

导数定义

导数和微分的概念

%3D%5Cunderset%7B%5CDelta%20x%5Cto%200%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf(%7B%7Bx%7D%7B0%7D%7D%2B%5CDelta%20x)-f(%7B%7Bx%7D%7B0%7D%7D)%7D%7B%5CDelta%20x%7D#card=math&code=f%27%28%7B%7Bx%7D%7B0%7D%7D%29%3D%5Cunderset%7B%5CDelta%20x%5Cto%200%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf%28%7B%7Bx%7D%7B0%7D%7D%2B%5CDelta%20x%29-f%28%7B%7Bx%7D_%7B0%7D%7D%29%7D%7B%5CDelta%20x%7D&id=WcEAT) （1）

或者：

%3D%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf(x)-f(%7B%7Bx%7D%7B0%7D%7D)%7D%7Bx-%7B%7Bx%7D%7B0%7D%7D%7D#card=math&code=f%27%28%7B%7Bx%7D%7B0%7D%7D%29%3D%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf%28x%29-f%28%7B%7Bx%7D%7B0%7D%7D%29%7D%7Bx-%7B%7Bx%7D_%7B0%7D%7D%7D&id=ya1wG) （2）

左右导数导数的几何意义和物理意义

函数#card=math&code=f%28x%29&id=B1Pio)在处的左、右导数分别定义为：

左导数：%3D%5Cunderset%7B%5CDelta%20x%5Cto%20%7B%7B0%7D%5E%7B-%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf(%7B%7Bx%7D%7B0%7D%7D%2B%5CDelta%20x)-f(%7B%7Bx%7D%7B0%7D%7D)%7D%7B%5CDelta%20x%7D%3D%5Cunderset%7Bx%5Cto%20x%7B0%7D%5E%7B-%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf(x)-f(%7B%7Bx%7D%7B0%7D%7D)%7D%7Bx-%7B%7Bx%7D%7B0%7D%7D%7D%2C(x%3D%7B%7Bx%7D%7B0%7D%7D%2B%5CDelta%20x)#card=math&code=%7B%7B%7Bf%7D%27%7D%7B-%7D%7D%28%7B%7Bx%7D%7B0%7D%7D%29%3D%5Cunderset%7B%5CDelta%20x%5Cto%20%7B%7B0%7D%5E%7B-%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf%28%7B%7Bx%7D%7B0%7D%7D%2B%5CDelta%20x%29-f%28%7B%7Bx%7D%7B0%7D%7D%29%7D%7B%5CDelta%20x%7D%3D%5Cunderset%7Bx%5Cto%20x%7B0%7D%5E%7B-%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf%28x%29-f%28%7B%7Bx%7D%7B0%7D%7D%29%7D%7Bx-%7B%7Bx%7D%7B0%7D%7D%7D%2C%28x%3D%7B%7Bx%7D%7B0%7D%7D%2B%5CDelta%20x%29&id=H4Qjx)

右导数：%3D%5Cunderset%7B%5CDelta%20x%5Cto%20%7B%7B0%7D%5E%7B%2B%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf(%7B%7Bx%7D%7B0%7D%7D%2B%5CDelta%20x)-f(%7B%7Bx%7D%7B0%7D%7D)%7D%7B%5CDelta%20x%7D%3D%5Cunderset%7Bx%5Cto%20x%7B0%7D%5E%7B%2B%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf(x)-f(%7B%7Bx%7D%7B0%7D%7D)%7D%7Bx-%7B%7Bx%7D%7B0%7D%7D%7D#card=math&code=%7B%7B%7Bf%7D%27%7D%7B%2B%7D%7D%28%7B%7Bx%7D%7B0%7D%7D%29%3D%5Cunderset%7B%5CDelta%20x%5Cto%20%7B%7B0%7D%5E%7B%2B%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf%28%7B%7Bx%7D%7B0%7D%7D%2B%5CDelta%20x%29-f%28%7B%7Bx%7D%7B0%7D%7D%29%7D%7B%5CDelta%20x%7D%3D%5Cunderset%7Bx%5Cto%20x%7B0%7D%5E%7B%2B%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf%28x%29-f%28%7B%7Bx%7D%7B0%7D%7D%29%7D%7Bx-%7B%7Bx%7D%7B0%7D%7D%7D&id=Uz89o)

函数的可导性与连续性之间的关系

Th1: 函数#card=math&code=f%28x%29&id=THtfG)在处可微#card=math&code=%5CLeftrightarrow%20f%28x%29&id=tQ43D)在处可导

Th2: 若函数在点处可导，则#card=math&code=y%3Df%28x%29&id=zaygb)在点处连续，反之则不成立。即函数连续不一定可导。

Th3: #card=math&code=%7Bf%7D%27%28%7B%7Bx%7D%7B0%7D%7D%29&id=wZUYV)存在%3D%7B%7B%7Bf%7D’%7D%7B%2B%7D%7D(%7B%7Bx%7D%7B0%7D%7D)#card=math&code=%5CLeftrightarrow%20%7B%7B%7Bf%7D%27%7D%7B-%7D%7D%28%7B%7Bx%7D%7B0%7D%7D%29%3D%7B%7B%7Bf%7D%27%7D%7B%2B%7D%7D%28%7B%7Bx%7D_%7B0%7D%7D%29&id=YuDpy)

平面曲线的切线和法线

切线方程 : (x-%7B%7Bx%7D%7B0%7D%7D)#card=math&code=y-%7B%7By%7D%7B0%7D%7D%3Df%27%28%7B%7Bx%7D%7B0%7D%7D%29%28x-%7B%7Bx%7D%7B0%7D%7D%29&id=hy71o)
法线方程：%7D(x-%7B%7Bx%7D%7B0%7D%7D)%2Cf’(%7B%7Bx%7D%7B0%7D%7D)%5Cne%200#card=math&code=y-%7B%7By%7D%7B0%7D%7D%3D-%5Cfrac%7B1%7D%7Bf%27%28%7B%7Bx%7D%7B0%7D%7D%29%7D%28x-%7B%7Bx%7D%7B0%7D%7D%29%2Cf%27%28%7B%7Bx%7D%7B0%7D%7D%29%5Cne%200&id=OJDB6)

四则运算法则

设函数%EF%BC%8Cv%3Dv(x)#card=math&code=u%3Du%28x%29%EF%BC%8Cv%3Dv%28x%29&id=ilnmp)]在点可导则
(1) %7D’%3D%7Bu%7D’%5Cpm%20%7Bv%7D’#card=math&code=%28u%5Cpm%20v%7B%29%7D%27%3D%7Bu%7D%27%5Cpm%20%7Bv%7D%27&id=NYu4E) %3Ddu%5Cpm%20dv#card=math&code=d%28u%5Cpm%20v%29%3Ddu%5Cpm%20dv&id=XsbIV)
(2)%7D’%3Du%7Bv%7D’%2Bv%7Bu%7D’#card=math&code=%28uv%7B%29%7D%27%3Du%7Bv%7D%27%2Bv%7Bu%7D%27&id=cm8Dw) %3Dudv%2Bvdu#card=math&code=d%28uv%29%3Dudv%2Bvdu&id=kGx4w)
(3) %7D’%3D%5Cfrac%7Bv%7Bu%7D’-u%7Bv%7D’%7D%7B%7B%7Bv%7D%5E%7B2%7D%7D%7D(v%5Cne%200)#card=math&code=%28%5Cfrac%7Bu%7D%7Bv%7D%7B%29%7D%27%3D%5Cfrac%7Bv%7Bu%7D%27-u%7Bv%7D%27%7D%7B%7B%7Bv%7D%5E%7B2%7D%7D%7D%28v%5Cne%200%29&id=TVU4Q) %3D%5Cfrac%7Bvdu-udv%7D%7B%7B%7Bv%7D%5E%7B2%7D%7D%7D#card=math&code=d%28%5Cfrac%7Bu%7D%7Bv%7D%29%3D%5Cfrac%7Bvdu-udv%7D%7B%7B%7Bv%7D%5E%7B2%7D%7D%7D&id=qy3Wt)

基本导数与微分表

(1) （常数）
(2) (为实数)
(3)
特例: %7D’%3D%7B%7B%7Be%7D%7D%5E%7Bx%7D%7D%E2%80%8B#card=math&code=%28%7B%7B%7Be%7D%7D%5E%7Bx%7D%7D%7B%29%7D%27%3D%7B%7B%7Be%7D%7D%5E%7Bx%7D%7D%E2%80%8B&id=nQwbe) %3D%7B%7B%7Be%7D%7D%5E%7Bx%7D%7Ddx%E2%80%8B#card=math&code=d%28%7B%7B%7Be%7D%7D%5E%7Bx%7D%7D%29%3D%7B%7B%7Be%7D%7D%5E%7Bx%7D%7Ddx%E2%80%8B&id=eNKXE)

(4)

特例: %7D’%3D%5Cfrac%7B1%7D%7Bx%7D#card=math&code=%28%5Cln%20x%7B%29%7D%27%3D%5Cfrac%7B1%7D%7Bx%7D&id=sGl1a) %3D%5Cfrac%7B1%7D%7Bx%7Ddx#card=math&code=d%28%5Cln%20x%29%3D%5Cfrac%7B1%7D%7Bx%7Ddx&id=hHvBB)

(5)

%3D%5Ccos%20xdx#card=math&code=d%28%5Csin%20x%29%3D%5Ccos%20xdx&id=WBZnW)

(6)

%3D-%5Csin%20xdx#card=math&code=d%28%5Ccos%20x%29%3D-%5Csin%20xdx&id=ln1Rs)

(7)

%3D%7B%7B%5Csec%20%7D%5E%7B2%7D%7Dxdx#card=math&code=d%28%5Ctan%20x%29%3D%7B%7B%5Csec%20%7D%5E%7B2%7D%7Dxdx&id=IBcco)
(8) %3D-%7B%7B%5Ccsc%20%7D%5E%7B2%7D%7Dxdx#card=math&code=d%28%5Ccot%20x%29%3D-%7B%7B%5Ccsc%20%7D%5E%7B2%7D%7Dxdx&id=NPpZX)
(9)

%3D%5Csec%20x%5Ctan%20xdx#card=math&code=d%28%5Csec%20x%29%3D%5Csec%20x%5Ctan%20xdx&id=dlNZ5)
(10)

%3D-%5Ccsc%20x%5Ccot%20xdx#card=math&code=d%28%5Ccsc%20x%29%3D-%5Ccsc%20x%5Ccot%20xdx&id=PtLMW)
(11)

%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B1-%7B%7Bx%7D%5E%7B2%7D%7D%7D%7Ddx#card=math&code=d%28%5Carcsin%20x%29%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B1-%7B%7Bx%7D%5E%7B2%7D%7D%7D%7Ddx&id=s5Bah)
(12)

%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7B1-%7B%7Bx%7D%5E%7B2%7D%7D%7D%7Ddx#card=math&code=d%28%5Carccos%20x%29%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7B1-%7B%7Bx%7D%5E%7B2%7D%7D%7D%7Ddx&id=nTIyt)

(13)

%3D%5Cfrac%7B1%7D%7B1%2B%7B%7Bx%7D%5E%7B2%7D%7D%7Ddx#card=math&code=d%28%5Carctan%20x%29%3D%5Cfrac%7B1%7D%7B1%2B%7B%7Bx%7D%5E%7B2%7D%7D%7Ddx&id=dllP4)

(14)

%3D-%5Cfrac%7B1%7D%7B1%2B%7B%7Bx%7D%5E%7B2%7D%7D%7Ddx#card=math&code=d%28%5Coperatorname%7Barc%7D%5Ccot%20x%29%3D-%5Cfrac%7B1%7D%7B1%2B%7B%7Bx%7D%5E%7B2%7D%7D%7Ddx&id=avOCE)
(15)

%3Dchxdx#card=math&code=d%28shx%29%3Dchxdx&id=ORfh5)

(16)

%3Dshxdx#card=math&code=d%28chx%29%3Dshxdx&id=Ojq9H)

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

(1) 反函数的运算法则: 设#card=math&code=y%3Df%28x%29&id=lEh7s)在点的某邻域内单调连续，在点处可导且%5Cne%200#card=math&code=%7Bf%7D%27%28x%29%5Cne%200&id=ItmIy)，则其反函数在点所对应的处可导，并且有
(2) 复合函数的运算法则:若#card=math&code=%5Cmu%20%3D%5Cvarphi%20%28x%29&id=HK7uM)在点可导,而#card=math&code=y%3Df%28%5Cmu%20%29&id=wqc4u)在对应点 ()可导,则复合函数在点可导,且
(3) 隐函数导数的求法一般有三种方法：
1)方程两边对求导，要记住是的函数，则的函数是的复合函数例如等均是的复合函数对求导应按复合函数连锁法则做.
2)公式法.由知 %7D%7B%7B%7B%7B%7BF%7D’%7D%7D%7By%7D%7D(x%2Cy)%7D#card=math&code=%5Cfrac%7Bdy%7D%7Bdx%7D%3D-%5Cfrac%7B%7B%7B%7B%7BF%7D%27%7D%7D%7Bx%7D%7D%28x%2Cy%29%7D%7B%7B%7B%7B%7BF%7D%27%7D%7D%7By%7D%7D%28x%2Cy%29%7D&id=HmyEY),其中，#card=math&code=%7B%7B%7BF%7D%27%7D%7Bx%7D%7D%28x%2Cy%29&id=Fk4Ga)，
#card=math&code=%7B%7B%7BF%7D%27%7D_%7By%7D%7D%28x%2Cy%29&id=tilau)分别表示#card=math&code=F%28x%2Cy%29&id=zG468)对和的偏导数
3)利用微分形式不变性

常用高阶导数公式

（1）%7B%7B%5C%2C%7D%5E%7B(n)%7D%7D%3D%7B%7Ba%7D%5E%7Bx%7D%7D%7B%7B%5Cln%20%7D%5E%7Bn%7D%7Da%5Cquad%20(a%3E%7B0%7D)%5Cquad%20%5Cquad%20(%7B%7B%7Be%7D%7D%5E%7Bx%7D%7D)%7B%7B%5C%2C%7D%5E%7B(n)%7D%7D%3D%7Be%7D%7B%7B%5C%2C%7D%5E%7Bx%7D%7D#card=math&code=%28%7B%7Ba%7D%5E%7Bx%7D%7D%29%7B%7B%5C%2C%7D%5E%7B%28n%29%7D%7D%3D%7B%7Ba%7D%5E%7Bx%7D%7D%7B%7B%5Cln%20%7D%5E%7Bn%7D%7Da%5Cquad%20%28a%3E%7B0%7D%29%5Cquad%20%5Cquad%20%28%7B%7B%7Be%7D%7D%5E%7Bx%7D%7D%29%7B%7B%5C%2C%7D%5E%7B%28n%29%7D%7D%3D%7Be%7D%7B%7B%5C%2C%7D%5E%7Bx%7D%7D&id=xdnH1)
（2）%7D%7B%7B%5C%2C%7D%5E%7B(n)%7D%7D%3D%7B%7Bk%7D%5E%7Bn%7D%7D%5Csin%20(kx%2Bn%5Ccdot%20%5Cfrac%7B%5Cpi%20%7D%7B%7B2%7D%7D)#card=math&code=%28%5Csin%20kx%7B%29%7D%7B%7B%5C%2C%7D%5E%7B%28n%29%7D%7D%3D%7B%7Bk%7D%5E%7Bn%7D%7D%5Csin%20%28kx%2Bn%5Ccdot%20%5Cfrac%7B%5Cpi%20%7D%7B%7B2%7D%7D%29&id=BUr7S)
（3）%7D%7B%7B%5C%2C%7D%5E%7B(n)%7D%7D%3D%7B%7Bk%7D%5E%7Bn%7D%7D%5Ccos%20(kx%2Bn%5Ccdot%20%5Cfrac%7B%5Cpi%20%7D%7B%7B2%7D%7D)#card=math&code=%28%5Ccos%20kx%7B%29%7D%7B%7B%5C%2C%7D%5E%7B%28n%29%7D%7D%3D%7B%7Bk%7D%5E%7Bn%7D%7D%5Ccos%20%28kx%2Bn%5Ccdot%20%5Cfrac%7B%5Cpi%20%7D%7B%7B2%7D%7D%29&id=xL6a6)
（4）%7B%7B%5C%2C%7D%5E%7B(n)%7D%7D%3Dm(m-1)%5Ccdots%20(m-n%2B1)%7B%7Bx%7D%5E%7Bm-n%7D%7D#card=math&code=%28%7B%7Bx%7D%5E%7Bm%7D%7D%29%7B%7B%5C%2C%7D%5E%7B%28n%29%7D%7D%3Dm%28m-1%29%5Ccdots%20%28m-n%2B1%29%7B%7Bx%7D%5E%7Bm-n%7D%7D&id=KBvW4)
（5）%7B%7B%5C%2C%7D%5E%7B(n)%7D%7D%3D%7B%7B(-%7B1%7D)%7D%5E%7B(n-%7B1%7D)%7D%7D%5Cfrac%7B(n-%7B1%7D)!%7D%7B%7B%7Bx%7D%5E%7Bn%7D%7D%7D#card=math&code=%28%5Cln%20x%29%7B%7B%5C%2C%7D%5E%7B%28n%29%7D%7D%3D%7B%7B%28-%7B1%7D%29%7D%5E%7B%28n-%7B1%7D%29%7D%7D%5Cfrac%7B%28n-%7B1%7D%29%21%7D%7B%7B%7Bx%7D%5E%7Bn%7D%7D%7D&id=uPSxD)
（6）莱布尼兹公式：若%5C%2C%2Cv(x)#card=math&code=u%28x%29%5C%2C%2Cv%28x%29&id=qGAkK)均阶可导，则
%7D%5E%7B(n)%7D%7D%3D%5Csum%5Climits%7Bi%3D%7B0%7D%7D%5E%7Bn%7D%7Bc%7Bn%7D%5E%7Bi%7D%7B%7Bu%7D%5E%7B(i)%7D%7D%7B%7Bv%7D%5E%7B(n-i)%7D%7D%7D#card=math&code=%7B%7B%28uv%29%7D%5E%7B%28n%29%7D%7D%3D%5Csum%5Climits%7Bi%3D%7B0%7D%7D%5E%7Bn%7D%7Bc%7Bn%7D%5E%7Bi%7D%7B%7Bu%7D%5E%7B%28i%29%7D%7D%7B%7Bv%7D%5E%7B%28n-i%29%7D%7D%7D&id=mUycT)，其中%7D%7D%3Du#card=math&code=%7B%7Bu%7D%5E%7B%28%7B0%7D%29%7D%7D%3Du&id=H677f)，%7D%7D%3Dv#card=math&code=%7B%7Bv%7D%5E%7B%28%7B0%7D%29%7D%7D%3Dv&id=jq62A)

微分中值定理，泰勒公式

Th1:(费马定理)

若函数#card=math&code=f%28x%29&id=F7wMM)满足条件：
(1)函数#card=math&code=f%28x%29&id=xiNpG)在的某邻域内有定义，并且在此邻域内恒有
%5Cle%20f(%7B%7Bx%7D%7B0%7D%7D)#card=math&code=f%28x%29%5Cle%20f%28%7B%7Bx%7D%7B0%7D%7D%29&id=y9agl)或%5Cge%20f(%7B%7Bx%7D%7B0%7D%7D)#card=math&code=f%28x%29%5Cge%20f%28%7B%7Bx%7D%7B0%7D%7D%29&id=rOwsd),

(2) #card=math&code=f%28x%29&id=s3mRs)在处可导,则有 %3D0#card=math&code=%7Bf%7D%27%28%7B%7Bx%7D_%7B0%7D%7D%29%3D0&id=rqCsq)

Th2:(罗尔定理)

设函数#card=math&code=f%28x%29&id=lQKfF)满足条件：
(1)在闭区间上连续；

(2)在#card=math&code=%28a%2Cb%29&id=UJynF)内可导；

(3)%3Df(b)#card=math&code=f%28a%29%3Df%28b%29&id=tlCSl)；

则在#card=math&code=%28a%2Cb%29&id=s3Cbt)内一存在个，使 %3D0#card=math&code=%7Bf%7D%27%28%5Cxi%20%29%3D0&id=UFCyg)
Th3: (拉格朗日中值定理)

设函数#card=math&code=f%28x%29&id=uSw0i)满足条件：
(1)在上连续；

(2)在#card=math&code=%28a%2Cb%29&id=AnkcS)内可导；

则在#card=math&code=%28a%2Cb%29&id=mfTG4)内一存在个 ，使 -f(a)%7D%7Bb-a%7D%3D%7Bf%7D’(%5Cxi%20)#card=math&code=%5Cfrac%7Bf%28b%29-f%28a%29%7D%7Bb-a%7D%3D%7Bf%7D%27%28%5Cxi%20%29&id=rZbPr)

Th4: (柯西中值定理)

设函数#card=math&code=f%28x%29&id=q8Sfg)，#card=math&code=g%28x%29&id=tWG1a)满足条件：
(1) 在上连续；

(2) 在#card=math&code=%28a%2Cb%29&id=eC9Dg)内可导且#card=math&code=%7Bf%7D%27%28x%29&id=rI1jz)，#card=math&code=%7Bg%7D%27%28x%29&id=riuhd)均存在，且%5Cne%200#card=math&code=%7Bg%7D%27%28x%29%5Cne%200&id=HNQiD)

则在#card=math&code=%28a%2Cb%29&id=MbJZI)内存在一个 ，使 -f(a)%7D%7Bg(b)-g(a)%7D%3D%5Cfrac%7B%7Bf%7D’(%5Cxi%20)%7D%7B%7Bg%7D’(%5Cxi%20)%7D#card=math&code=%5Cfrac%7Bf%28b%29-f%28a%29%7D%7Bg%28b%29-g%28a%29%7D%3D%5Cfrac%7B%7Bf%7D%27%28%5Cxi%20%29%7D%7B%7Bg%7D%27%28%5Cxi%20%29%7D&id=lwsov)

洛必达法则

法则Ⅰ (型)
设函数%2Cg%5Cleft(%20x%20%5Cright)#card=math&code=f%5Cleft%28%20x%20%5Cright%29%2Cg%5Cleft%28%20x%20%5Cright%29&id=eiO5Y)满足条件：
%3D0%2C%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2Cg%5Cleft(%20x%20%5Cright)%3D0#card=math&code=%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2Cf%5Cleft%28%20x%20%5Cright%29%3D0%2C%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D_%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2Cg%5Cleft%28%20x%20%5Cright%29%3D0&id=KTs5o);

%2Cg%5Cleft(%20x%20%5Cright)#card=math&code=f%5Cleft%28%20x%20%5Cright%29%2Cg%5Cleft%28%20x%20%5Cright%29&id=H9Fhn)在的邻域内可导，(在处可除外)且%5Cne%200#card=math&code=%7Bg%7D%27%5Cleft%28%20x%20%5Cright%29%5Cne%200&id=zqtxq);

%7D%7B%7Bg%7D’%5Cleft(%20x%20%5Cright)%7D#card=math&code=%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D_%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7B%7Bf%7D%27%5Cleft%28%20x%20%5Cright%29%7D%7B%7Bg%7D%27%5Cleft%28%20x%20%5Cright%29%7D&id=lzacO)存在(或 )。

则:
%7D%7Bg%5Cleft(%20x%20%5Cright)%7D%3D%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7B%7Bf%7D’%5Cleft(%20x%20%5Cright)%7D%7B%7Bg%7D’%5Cleft(%20x%20%5Cright)%7D#card=math&code=%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf%5Cleft%28%20x%20%5Cright%29%7D%7Bg%5Cleft%28%20x%20%5Cright%29%7D%3D%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D_%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7B%7Bf%7D%27%5Cleft%28%20x%20%5Cright%29%7D%7B%7Bg%7D%27%5Cleft%28%20x%20%5Cright%29%7D&id=SVW4q)。
法则 (型)设函数%2Cg%5Cleft(%20x%20%5Cright)#card=math&code=f%5Cleft%28%20x%20%5Cright%29%2Cg%5Cleft%28%20x%20%5Cright%29&id=PvoDV)满足条件：
%3D0%2C%5Cunderset%7Bx%5Cto%20%5Cinfty%20%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2Cg%5Cleft(%20x%20%5Cright)%3D0#card=math&code=%5Cunderset%7Bx%5Cto%20%5Cinfty%20%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2Cf%5Cleft%28%20x%20%5Cright%29%3D0%2C%5Cunderset%7Bx%5Cto%20%5Cinfty%20%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2Cg%5Cleft%28%20x%20%5Cright%29%3D0&id=Tsqba);

存在一个,当时,%2Cg%5Cleft(%20x%20%5Cright)#card=math&code=f%5Cleft%28%20x%20%5Cright%29%2Cg%5Cleft%28%20x%20%5Cright%29&id=dNxvB)可导,且%5Cne%200#card=math&code=%7Bg%7D%27%5Cleft%28%20x%20%5Cright%29%5Cne%200&id=bG4x7);%7D%7B%7Bg%7D’%5Cleft(%20x%20%5Cright)%7D#card=math&code=%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D_%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7B%7Bf%7D%27%5Cleft%28%20x%20%5Cright%29%7D%7B%7Bg%7D%27%5Cleft%28%20x%20%5Cright%29%7D&id=CuK9H)存在(或$\infty $)。

则:
%7D%7Bg%5Cleft(%20x%20%5Cright)%7D%3D%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7B%7Bf%7D’%5Cleft(%20x%20%5Cright)%7D%7B%7Bg%7D’%5Cleft(%20x%20%5Cright)%7D#card=math&code=%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf%5Cleft%28%20x%20%5Cright%29%7D%7Bg%5Cleft%28%20x%20%5Cright%29%7D%3D%5Cunderset%7Bx%5Cto%20%7B%7Bx%7D%7B0%7D%7D%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7B%7Bf%7D%27%5Cleft%28%20x%20%5Cright%29%7D%7B%7Bg%7D%27%5Cleft%28%20x%20%5Cright%29%7D&id=WxqfG)
法则Ⅱ(型) 设函数%2Cg%5Cleft(%20x%20%5Cright)#card=math&code=f%5Cleft%28%20x%20%5Cright%29%2Cg%5Cleft%28%20x%20%5Cright%29&id=WnIzo)满足条件：
; 在邻域内可导(在处可除外）且；存在（或）。则同理法则（型）仿法则可写出。

泰勒公式

设函数#card=math&code=f%28x%29&id=l4nHq)在点处的某邻域内具有阶导数，则对该邻域内异于的任意点，在与之间至少存在
一个
%7D%7D(%7B%7Bx%7D%7B0%7D%7D)%7D%7Bn!%7D%7B%7B(x-%7B%7Bx%7D%7B0%7D%7D)%7D%5E%7Bn%7D%7D%2B%7B%7BR%7D%7Bn%7D%7D(x)#card=math&code=%2B%5Cfrac%7B%7B%7Bf%7D%5E%7B%28n%29%7D%7D%28%7B%7Bx%7D%7B0%7D%7D%29%7D%7Bn%21%7D%7B%7B%28x-%7B%7Bx%7D%7B0%7D%7D%29%7D%5E%7Bn%7D%7D%2B%7B%7BR%7D%7Bn%7D%7D%28x%29&id=zulXz)
其中 %3D%5Cfrac%7B%7B%7Bf%7D%5E%7B(n%2B1)%7D%7D(%5Cxi%20)%7D%7B(n%2B1)!%7D%7B%7B(x-%7B%7Bx%7D%7B0%7D%7D)%7D%5E%7Bn%2B1%7D%7D#card=math&code=%7B%7BR%7D%7Bn%7D%7D%28x%29%3D%5Cfrac%7B%7B%7Bf%7D%5E%7B%28n%2B1%29%7D%7D%28%5Cxi%20%29%7D%7B%28n%2B1%29%21%7D%7B%7B%28x-%7B%7Bx%7D%7B0%7D%7D%29%7D%5E%7Bn%2B1%7D%7D&id=BzEEC)称为#card=math&code=f%28x%29&id=cWHbk)在点处的阶泰勒余项。

令，则阶泰勒公式
%3Df(0)%2B%7Bf%7D’(0)x%2B%5Cfrac%7B1%7D%7B2!%7D%7Bf%7D’’(0)%7B%7Bx%7D%5E%7B2%7D%7D%2B%5Ccdots%20%2B%5Cfrac%7B%7B%7Bf%7D%5E%7B(n)%7D%7D(0)%7D%7Bn!%7D%7B%7Bx%7D%5E%7Bn%7D%7D%2B%7B%7BR%7D%7Bn%7D%7D(x)#card=math&code=f%28x%29%3Df%280%29%2B%7Bf%7D%27%280%29x%2B%5Cfrac%7B1%7D%7B2%21%7D%7Bf%7D%27%27%280%29%7B%7Bx%7D%5E%7B2%7D%7D%2B%5Ccdots%20%2B%5Cfrac%7B%7B%7Bf%7D%5E%7B%28n%29%7D%7D%280%29%7D%7Bn%21%7D%7B%7Bx%7D%5E%7Bn%7D%7D%2B%7B%7BR%7D%7Bn%7D%7D%28x%29&id=LNaOp)……(1)
其中 %3D%5Cfrac%7B%7B%7Bf%7D%5E%7B(n%2B1)%7D%7D(%5Cxi%20)%7D%7B(n%2B1)!%7D%7B%7Bx%7D%5E%7Bn%2B1%7D%7D#card=math&code=%7B%7BR%7D_%7Bn%7D%7D%28x%29%3D%5Cfrac%7B%7B%7Bf%7D%5E%7B%28n%2B1%29%7D%7D%28%5Cxi%20%29%7D%7B%28n%2B1%29%21%7D%7B%7Bx%7D%5E%7Bn%2B1%7D%7D&id=vXsnv)， 在 0 与 之间.(1)式称为麦克劳林公式

常用五种函数在处的泰勒公式

(1) !%7D%7B%7Be%7D%5E%7B%5Cxi%20%7D%7D#card=math&code=%7B%7B%7Be%7D%7D%5E%7Bx%7D%7D%3D1%2Bx%2B%5Cfrac%7B1%7D%7B2%21%7D%7B%7Bx%7D%5E%7B2%7D%7D%2B%5Ccdots%20%2B%5Cfrac%7B1%7D%7Bn%21%7D%7B%7Bx%7D%5E%7Bn%7D%7D%2B%5Cfrac%7B%7B%7Bx%7D%5E%7Bn%2B1%7D%7D%7D%7B%28n%2B1%29%21%7D%7B%7Be%7D%5E%7B%5Cxi%20%7D%7D&id=GVwIt)

或 #card=math&code=%3D1%2Bx%2B%5Cfrac%7B1%7D%7B2%21%7D%7B%7Bx%7D%5E%7B2%7D%7D%2B%5Ccdots%20%2B%5Cfrac%7B1%7D%7Bn%21%7D%7B%7Bx%7D%5E%7Bn%7D%7D%2Bo%28%7B%7Bx%7D%5E%7Bn%7D%7D%29&id=XkIHH)

(2) !%7D%5Csin%20(%5Cxi%20%2B%5Cfrac%7Bn%2B1%7D%7B2%7D%5Cpi%20)#card=math&code=%5Csin%20x%3Dx-%5Cfrac%7B1%7D%7B3%21%7D%7B%7Bx%7D%5E%7B3%7D%7D%2B%5Ccdots%20%2B%5Cfrac%7B%7B%7Bx%7D%5E%7Bn%7D%7D%7D%7Bn%21%7D%5Csin%20%5Cfrac%7Bn%5Cpi%20%7D%7B2%7D%2B%5Cfrac%7B%7B%7Bx%7D%5E%7Bn%2B1%7D%7D%7D%7B%28n%2B1%29%21%7D%5Csin%20%28%5Cxi%20%2B%5Cfrac%7Bn%2B1%7D%7B2%7D%5Cpi%20%29&id=GEIZo)

或 #card=math&code=%3Dx-%5Cfrac%7B1%7D%7B3%21%7D%7B%7Bx%7D%5E%7B3%7D%7D%2B%5Ccdots%20%2B%5Cfrac%7B%7B%7Bx%7D%5E%7Bn%7D%7D%7D%7Bn%21%7D%5Csin%20%5Cfrac%7Bn%5Cpi%20%7D%7B2%7D%2Bo%28%7B%7Bx%7D%5E%7Bn%7D%7D%29&id=uKMbo)

(3) !%7D%5Ccos%20(%5Cxi%20%2B%5Cfrac%7Bn%2B1%7D%7B2%7D%5Cpi%20)#card=math&code=%5Ccos%20x%3D1-%5Cfrac%7B1%7D%7B2%21%7D%7B%7Bx%7D%5E%7B2%7D%7D%2B%5Ccdots%20%2B%5Cfrac%7B%7B%7Bx%7D%5E%7Bn%7D%7D%7D%7Bn%21%7D%5Ccos%20%5Cfrac%7Bn%5Cpi%20%7D%7B2%7D%2B%5Cfrac%7B%7B%7Bx%7D%5E%7Bn%2B1%7D%7D%7D%7B%28n%2B1%29%21%7D%5Ccos%20%28%5Cxi%20%2B%5Cfrac%7Bn%2B1%7D%7B2%7D%5Cpi%20%29&id=NKAEA)

或 #card=math&code=%3D1-%5Cfrac%7B1%7D%7B2%21%7D%7B%7Bx%7D%5E%7B2%7D%7D%2B%5Ccdots%20%2B%5Cfrac%7B%7B%7Bx%7D%5E%7Bn%7D%7D%7D%7Bn%21%7D%5Ccos%20%5Cfrac%7Bn%5Cpi%20%7D%7B2%7D%2Bo%28%7B%7Bx%7D%5E%7Bn%7D%7D%29&id=E6KB0)

(4) %3Dx-%5Cfrac%7B1%7D%7B2%7D%7B%7Bx%7D%5E%7B2%7D%7D%2B%5Cfrac%7B1%7D%7B3%7D%7B%7Bx%7D%5E%7B3%7D%7D-%5Ccdots%20%2B%7B%7B(-1)%7D%5E%7Bn-1%7D%7D%5Cfrac%7B%7B%7Bx%7D%5E%7Bn%7D%7D%7D%7Bn%7D%2B%5Cfrac%7B%7B%7B(-1)%7D%5E%7Bn%7D%7D%7B%7Bx%7D%5E%7Bn%2B1%7D%7D%7D%7B(n%2B1)%7B%7B(1%2B%5Cxi%20)%7D%5E%7Bn%2B1%7D%7D%7D#card=math&code=%5Cln%20%281%2Bx%29%3Dx-%5Cfrac%7B1%7D%7B2%7D%7B%7Bx%7D%5E%7B2%7D%7D%2B%5Cfrac%7B1%7D%7B3%7D%7B%7Bx%7D%5E%7B3%7D%7D-%5Ccdots%20%2B%7B%7B%28-1%29%7D%5E%7Bn-1%7D%7D%5Cfrac%7B%7B%7Bx%7D%5E%7Bn%7D%7D%7D%7Bn%7D%2B%5Cfrac%7B%7B%7B%28-1%29%7D%5E%7Bn%7D%7D%7B%7Bx%7D%5E%7Bn%2B1%7D%7D%7D%7B%28n%2B1%29%7B%7B%281%2B%5Cxi%20%29%7D%5E%7Bn%2B1%7D%7D%7D&id=WGCGk)

或 %7D%5E%7Bn-1%7D%7D%5Cfrac%7B%7B%7Bx%7D%5E%7Bn%7D%7D%7D%7Bn%7D%2Bo(%7B%7Bx%7D%5E%7Bn%7D%7D)#card=math&code=%3Dx-%5Cfrac%7B1%7D%7B2%7D%7B%7Bx%7D%5E%7B2%7D%7D%2B%5Cfrac%7B1%7D%7B3%7D%7B%7Bx%7D%5E%7B3%7D%7D-%5Ccdots%20%2B%7B%7B%28-1%29%7D%5E%7Bn-1%7D%7D%5Cfrac%7B%7B%7Bx%7D%5E%7Bn%7D%7D%7D%7Bn%7D%2Bo%28%7B%7Bx%7D%5E%7Bn%7D%7D%29&id=ib1BR)

(5) %7D%5E%7Bm%7D%7D%3D1%2Bmx%2B%5Cfrac%7Bm(m-1)%7D%7B2!%7D%7B%7Bx%7D%5E%7B2%7D%7D%2B%5Ccdots%20%2B%5Cfrac%7Bm(m-1)%5Ccdots%20(m-n%2B1)%7D%7Bn!%7D%7B%7Bx%7D%5E%7Bn%7D%7D#card=math&code=%7B%7B%281%2Bx%29%7D%5E%7Bm%7D%7D%3D1%2Bmx%2B%5Cfrac%7Bm%28m-1%29%7D%7B2%21%7D%7B%7Bx%7D%5E%7B2%7D%7D%2B%5Ccdots%20%2B%5Cfrac%7Bm%28m-1%29%5Ccdots%20%28m-n%2B1%29%7D%7Bn%21%7D%7B%7Bx%7D%5E%7Bn%7D%7D&id=dwmZs)
%5Ccdots%20(m-n%2B1)%7D%7B(n%2B1)!%7D%7B%7Bx%7D%5E%7Bn%2B1%7D%7D%7B%7B(1%2B%5Cxi%20)%7D%5E%7Bm-n-1%7D%7D#card=math&code=%2B%5Cfrac%7Bm%28m-1%29%5Ccdots%20%28m-n%2B1%29%7D%7B%28n%2B1%29%21%7D%7B%7Bx%7D%5E%7Bn%2B1%7D%7D%7B%7B%281%2B%5Cxi%20%29%7D%5E%7Bm-n-1%7D%7D&id=JAz0e)

或 %5Ccdots%20(m-n%2B1)%7D%7Bn!%7D%7B%7Bx%7D%5E%7Bn%7D%7D%2Bo(%7B%7Bx%7D%5E%7Bn%7D%7D)#card=math&code=%2B%5Cfrac%7Bm%28m-1%29%5Ccdots%20%28m-n%2B1%29%7D%7Bn%21%7D%7B%7Bx%7D%5E%7Bn%7D%7D%2Bo%28%7B%7Bx%7D%5E%7Bn%7D%7D%29&id=jpNb9)

函数单调性的判断

Th1: 设函数#card=math&code=f%28x%29&id=Fsq5M)在#card=math&code=%28a%2Cb%29&id=IU6zl)区间内可导，如果对#card=math&code=%5Cforall%20x%5Cin%20%28a%2Cb%29&id=jaRHl)，都有%3E0#card=math&code=f%5C%2C%27%28x%29%3E0&id=yjtmQ)（或%3C0#card=math&code=f%5C%2C%27%28x%29%3C0&id=s02i5)），则函数#card=math&code=f%28x%29&id=u2FhU)在#card=math&code=%28a%2Cb%29&id=q1JCm)内是单调增加的（或单调减少）

Th2: （取极值的必要条件）设函数#card=math&code=f%28x%29&id=Pa5fJ)在处可导，且在处取极值，则%3D0#card=math&code=f%5C%2C%27%28%7B%7Bx%7D_%7B0%7D%7D%29%3D0&id=Rc8L3)。

Th3: （取极值的第一充分条件）设函数#card=math&code=f%28x%29&id=JaehT)在的某一邻域内可微，且%3D0#card=math&code=f%5C%2C%27%28%7B%7Bx%7D%7B0%7D%7D%29%3D0&id=CLtGa)（或#card=math&code=f%28x%29&id=thCvL)在处连续，但#card=math&code=f%5C%2C%27%28%7B%7Bx%7D%7B0%7D%7D%29&id=NySIp)不存在。）
(1)若当经过时，#card=math&code=f%5C%2C%27%28x%29&id=iSpSI)由“+”变“-”，则#card=math&code=f%28%7B%7Bx%7D%7B0%7D%7D%29&id=MuXs4)为极大值；
(2)若当经过时，#card=math&code=f%5C%2C%27%28x%29&id=oInBo)由“-”变“+”，则#card=math&code=f%28%7B%7Bx%7D%7B0%7D%7D%29&id=PuKdV)为极小值；
(3)若#card=math&code=f%5C%2C%27%28x%29&id=zuSpy)经过的两侧不变号，则#card=math&code=f%28%7B%7Bx%7D_%7B0%7D%7D%29&id=jfJHG)不是极值。

Th4: (取极值的第二充分条件)设#card=math&code=f%28x%29&id=NPkLJ)在点处有%5Cne%200#card=math&code=f%27%27%28x%29%5Cne%200&id=IK1iL)，且%3D0#card=math&code=f%5C%2C%27%28%7B%7Bx%7D%7B0%7D%7D%29%3D0&id=o2tHC)，则 当%3C0#card=math&code=f%27%5C%2C%27%28%7B%7Bx%7D%7B0%7D%7D%29%3C0&id=Ut1gi)时，#card=math&code=f%28%7B%7Bx%7D%7B0%7D%7D%29&id=jAS8U)为极大值；
当%3E0#card=math&code=f%27%5C%2C%27%28%7B%7Bx%7D%7B0%7D%7D%29%3E0&id=PCHx1)时，#card=math&code=f%28%7B%7Bx%7D%7B0%7D%7D%29&id=qRHBP)为极小值。
注：如果%3C0#card=math&code=f%27%5C%2C%27%28%7B%7Bx%7D_%7B0%7D%7D%29%3C0&id=HLThx)，此方法失效。

渐近线的求法

(1)水平渐近线 若%3Db#card=math&code=%5Cunderset%7Bx%5Cto%20%2B%5Cinfty%20%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2Cf%28x%29%3Db&id=aHcf2)，或%3Db#card=math&code=%5Cunderset%7Bx%5Cto%20-%5Cinfty%20%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2Cf%28x%29%3Db&id=RRVNo)，则

称为函数#card=math&code=y%3Df%28x%29&id=PWVpw)的水平渐近线。

(2)铅直渐近线 若 或 ，则

称为#card=math&code=y%3Df%28x%29&id=wvg18)的铅直渐近线。

(3)斜渐近线 若%7D%7Bx%7D%2C%5Cquad%20b%3D%5Cunderset%7Bx%5Cto%20%5Cinfty%20%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Bf(x)-ax%5D#card=math&code=a%3D%5Cunderset%7Bx%5Cto%20%5Cinfty%20%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Cfrac%7Bf%28x%29%7D%7Bx%7D%2C%5Cquad%20b%3D%5Cunderset%7Bx%5Cto%20%5Cinfty%20%7D%7B%5Cmathop%7B%5Clim%20%7D%7D%5C%2C%5Bf%28x%29-ax%5D&id=UNE5P)，则
称为#card=math&code=y%3Df%28x%29&id=bZUft)的斜渐近线。

函数凹凸性的判断

Th1: (凹凸性的判别定理）若在I上%3C0#card=math&code=f%27%27%28x%29%3C0&id=q9Mci)（或%3E0#card=math&code=f%27%27%28x%29%3E0&id=KLVgx)），则#card=math&code=f%28x%29&id=aTJNc)在I上是凸的（或凹的）。

Th2: (拐点的判别定理1)若在处%3D0#card=math&code=f%27%27%28x%29%3D0&id=JZIil)，（或#card=math&code=f%27%27%28x%29&id=GpDqf)不存在），当变动经过时，#card=math&code=f%27%27%28x%29&id=RwiW7)变号，则)#card=math&code=%28%7B%7Bx%7D%7B0%7D%7D%2Cf%28%7B%7Bx%7D%7B0%7D%7D%29%29&id=J2Aoa)为拐点。

Th3: (拐点的判别定理2)设#card=math&code=f%28x%29&id=XS4ZI)在点的某邻域内有三阶导数，且%3D0#card=math&code=f%27%27%28x%29%3D0&id=wzmTG)，%5Cne%200#card=math&code=f%27%27%27%28x%29%5Cne%200&id=jnGp9)，则)#card=math&code=%28%7B%7Bx%7D%7B0%7D%7D%2Cf%28%7B%7Bx%7D%7B0%7D%7D%29%29&id=GBZA0)为拐点。

弧微分

曲率

曲线#card=math&code=y%3Df%28x%29&id=wN0Kb)在点#card=math&code=%28x%2Cy%29&id=Zuk7d)处的曲率%7D%5E%7B%5Ctfrac%7B3%7D%7B2%7D%7D%7D%7D#card=math&code=k%3D%5Cfrac%7B%5Cleft%7C%20y%27%27%20%5Cright%7C%7D%7B%7B%7B%281%2By%7B%7B%27%7D%5E%7B2%7D%7D%29%7D%5E%7B%5Ctfrac%7B3%7D%7B2%7D%7D%7D%7D&id=oKlTw)。
对于参数方程%20%5C%5C%20%20%26%20y%3D%5Cpsi%20(t)%20%5C%5C%20%5Cend%7Balign%7D%20%5Cright.%2C#card=math&code=%5Cleft%5C%7B%20%5Cbegin%7Balign%7D%20%20%26%20x%3D%5Cvarphi%20%28t%29%20%5C%5C%20%20%26%20y%3D%5Cpsi%20%28t%29%20%5C%5C%20%5Cend%7Balign%7D%20%5Cright.%2C&id=dd7wt)%5Cpsi%20’’(t)-%5Cvarphi%20’’(t)%5Cpsi%20’(t)%20%5Cright%7C%7D%7B%7B%7B%5B%5Cvarphi%20%7B%7B’%7D%5E%7B2%7D%7D(t)%2B%5Cpsi%20%7B%7B’%7D%5E%7B2%7D%7D(t)%5D%7D%5E%7B%5Ctfrac%7B3%7D%7B2%7D%7D%7D%7D#card=math&code=k%3D%5Cfrac%7B%5Cleft%7C%20%5Cvarphi%20%27%28t%29%5Cpsi%20%27%27%28t%29-%5Cvarphi%20%27%27%28t%29%5Cpsi%20%27%28t%29%20%5Cright%7C%7D%7B%7B%7B%5B%5Cvarphi%20%7B%7B%27%7D%5E%7B2%7D%7D%28t%29%2B%5Cpsi%20%7B%7B%27%7D%5E%7B2%7D%7D%28t%29%5D%7D%5E%7B%5Ctfrac%7B3%7D%7B2%7D%7D%7D%7D&id=UrZ3q)。

曲率半径

曲线在点处的曲率#card=math&code=k%28k%5Cne%200%29&id=qI87K)与曲线在点处的曲率半径 。

线性代数

行列式

1.行列式按行（列）展开定理

(1) 设%7Bn%20%5Ctimes%20n%7D#card=math&code=A%20%3D%20%28%20a%7B%7Bij%7D%7D%20%29%7Bn%20%5Ctimes%20n%7D&id=yGxWE)，则：

或即 其中：%20%3D%20%7B(A%7B%7Bij%7D%7D)%7D%5E%7BT%7D#card=math&code=A%5E%7B%2A%7D%20%3D%20%5Cbegin%7Bpmatrix%7D%20A%7B11%7D%20%26%20A%7B12%7D%20%26%20%5Cldots%20%26%20A%7B1n%7D%20%5C%5C%20A%7B21%7D%20%26%20A%7B22%7D%20%26%20%5Cldots%20%26%20A%7B2n%7D%20%5C%5C%20%5Cldots%20%26%20%5Cldots%20%26%20%5Cldots%20%26%20%5Cldots%20%5C%5C%20A%7Bn1%7D%20%26%20A%7Bn2%7D%20%26%20%5Cldots%20%26%20A%7B%7Bnn%7D%7D%20%5C%5C%20%5Cend%7Bpmatrix%7D%20%3D%20%28A%7B%7Bji%7D%7D%29%20%3D%20%7B%28A%7B%7Bij%7D%7D%29%7D%5E%7BT%7D&id=DN3EJ)

#card=math&code=D%7Bn%7D%20%3D%20%5Cbegin%7Bvmatrix%7D%201%20%26%201%20%26%20%5Cldots%20%26%201%20%5C%5C%20x%7B1%7D%20%26%20x%7B2%7D%20%26%20%5Cldots%20%26%20x%7Bn%7D%20%5C%5C%20%5Cldots%20%26%20%5Cldots%20%26%20%5Cldots%20%26%20%5Cldots%20%5C%5C%20x%7B1%7D%5E%7Bn%20-%201%7D%20%26%20x%7B2%7D%5E%7Bn%20-%201%7D%20%26%20%5Cldots%20%26%20x%7Bn%7D%5E%7Bn%20-%201%7D%20%5C%5C%20%5Cend%7Bvmatrix%7D%20%3D%20%5Cprod%7B1%20%5Cleq%20j%20%3C%20i%20%5Cleq%20n%7D%5E%7B%7D%5C%2C%28x%7Bi%7D%20-%20x%7Bj%7D%29&id=Ho706)

(2) 设为阶方阵，则，但不一定成立。

(3) ,为阶方阵。

(4) 设为阶方阵，（若可逆），

(5)
，为方阵，但%7D%5E%7B%7Bmn%7D%7D%7CA%7C%7CB%7C#card=math&code=%5Cleft%7C%20%5Cbegin%7Bmatrix%7D%20%7BO%7D%20%26%20A%7Bm%20%5Ctimes%20m%7D%20%5C%5C%20%20B%7Bn%20%5Ctimes%20n%7D%20%26%20%7B%20O%7D%20%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%7C%20%3D%20%28%7B-%201%29%7D%5E%7B%7Bmn%7D%7D%7CA%7C%7CB%7C&id=tWtfm) 。

(6) 范德蒙行列式#card=math&code=D%7Bn%7D%20%3D%20%5Cbegin%7Bvmatrix%7D%201%20%26%201%20%26%20%5Cldots%20%26%201%20%5C%5C%20x%7B1%7D%20%26%20x%7B2%7D%20%26%20%5Cldots%20%26%20x%7Bn%7D%20%5C%5C%20%5Cldots%20%26%20%5Cldots%20%26%20%5Cldots%20%26%20%5Cldots%20%5C%5C%20x%7B1%7D%5E%7Bn%20-%201%7D%20%26%20x%7B2%7D%5E%7Bn%201%7D%20%26%20%5Cldots%20%26%20x%7Bn%7D%5E%7Bn%20-%201%7D%20%5C%5C%20%5Cend%7Bvmatrix%7D%20%3D%20%20%5Cprod%7B1%20%5Cleq%20j%20%3C%20i%20%5Cleq%20n%7D%5E%7B%7D%5C%2C%28x%7Bi%7D%20-%20x%7Bj%7D%29&id=QgkiP)

设是阶方阵，#card=math&code=%5Clambda%7Bi%7D%28i%20%3D%201%2C2%5Ccdots%2Cn%29&id=vfJst)是的个特征值，则

矩阵

矩阵：个数排成行列的表格 称为矩阵，简记为，或者%7Bm%20%5Ctimes%20n%7D#card=math&code=%5Cleft%28%20a%7B%7Bij%7D%7D%20%5Cright%29_%7Bm%20%5Ctimes%20n%7D&id=utgyD) 。若，则称是阶矩阵或阶方阵。

矩阵的线性运算

1.矩阵的加法

设%2CB%20%3D%20(b%7B%7Bij%7D%7D)#card=math&code=A%20%3D%20%28a%7B%7Bij%7D%7D%29%2CB%20%3D%20%28b%7B%7Bij%7D%7D%29&id=XlQXJ)是两个矩阵，则 矩阵%20%3D%20a%7B%7Bij%7D%7D%20%2B%20b%7B%7Bij%7D%7D#card=math&code=C%20%3D%20c%7B%7Bij%7D%7D%29%20%3D%20a%7B%7Bij%7D%7D%20%2B%20b_%7B%7Bij%7D%7D&id=sHFdx)称为矩阵与的和，记为 。

2.矩阵的数乘

设#card=math&code=A%20%3D%20%28a%7B%7Bij%7D%7D%29&id=pnA2t)是矩阵，是一个常数，则矩阵#card=math&code=%28ka_%7B%7Bij%7D%7D%29&id=A3Qq3)称为数与矩阵的数乘，记为。

3.矩阵的乘法

设#card=math&code=A%20%3D%20%28a%7B%7Bij%7D%7D%29&id=ONIsM)是矩阵，#card=math&code=B%20%3D%20%28b%7B%7Bij%7D%7D%29&id=jvBz5)是矩阵，那么矩阵#card=math&code=C%20%3D%20%28c%7B%7Bij%7D%7D%29&id=oPF4f)，其中称为的乘积，记为 。

4. 、、三者之间的关系

(1) %7D%5E%7BT%7D%20%3D%20A%2C%7B(AB)%7D%5E%7BT%7D%20%3D%20B%5E%7BT%7DA%5E%7BT%7D%2C%7B(kA)%7D%5E%7BT%7D%20%3D%20kA%5E%7BT%7D%2C%7B(A%20%5Cpm%20B)%7D%5E%7BT%7D%20%3D%20A%5E%7BT%7D%20%5Cpm%20B%5E%7BT%7D#card=math&code=%7B%28A%5E%7BT%7D%29%7D%5E%7BT%7D%20%3D%20A%2C%7B%28AB%29%7D%5E%7BT%7D%20%3D%20B%5E%7BT%7DA%5E%7BT%7D%2C%7B%28kA%29%7D%5E%7BT%7D%20%3D%20kA%5E%7BT%7D%2C%7B%28A%20%5Cpm%20B%29%7D%5E%7BT%7D%20%3D%20A%5E%7BT%7D%20%5Cpm%20B%5E%7BT%7D&id=CSvRX)

(2) %5E%7B-%201%7D%20%3D%20A%2C%5Cleft(%20%7BAB%7D%20%5Cright)%5E%7B-%201%7D%20%3D%20B%5E%7B-%201%7DA%5E%7B-%201%7D%2C%5Cleft(%20%7BkA%7D%20%5Cright)%5E%7B-%201%7D%20%3D%20%5Cfrac%7B1%7D%7Bk%7DA%5E%7B-%201%7D%2C#card=math&code=%5Cleft%28%20A%5E%7B-%201%7D%20%5Cright%29%5E%7B-%201%7D%20%3D%20A%2C%5Cleft%28%20%7BAB%7D%20%5Cright%29%5E%7B-%201%7D%20%3D%20B%5E%7B-%201%7DA%5E%7B-%201%7D%2C%5Cleft%28%20%7BkA%7D%20%5Cright%29%5E%7B-%201%7D%20%3D%20%5Cfrac%7B1%7D%7Bk%7DA%5E%7B-%201%7D%2C&id=gMSPj)

但 %7D%5E%7B-%201%7D%20%3D%20A%5E%7B-%201%7D%20%5Cpm%20B%5E%7B-%201%7D#card=math&code=%7B%28A%20%5Cpm%20B%29%7D%5E%7B-%201%7D%20%3D%20A%5E%7B-%201%7D%20%5Cpm%20B%5E%7B-%201%7D&id=svfsW)不一定成立。

(3) %5E%7B%7D%20%3D%20%7CA%7C%5E%7Bn%20-%202%7D%5C%20A%5C%20%5C%20(n%20%5Cgeq%203)#card=math&code=%5Cleft%28%20A%5E%7B%2A%7D%20%5Cright%29%5E%7B%2A%7D%20%3D%20%7CA%7C%5E%7Bn%20-%202%7D%5C%20A%5C%20%5C%20%28n%20%5Cgeq%203%29&id=HsWGm)，%5E%7B%7D%20%3D%20B%5E%7B%7DA%5E%7B%7D%2C#card=math&code=%5Cleft%28%7BAB%7D%20%5Cright%29%5E%7B%2A%7D%20%3D%20B%5E%7B%2A%7DA%5E%7B%2A%7D%2C&id=xhE1U) %5E%7B%7D%20%3D%20k%5E%7Bn%20-1%7DA%5E%7B%7D%7B%5C%20%5C%20%7D%5Cleft(%20n%20%5Cgeq%202%20%5Cright)#card=math&code=%5Cleft%28%20%7BkA%7D%20%5Cright%29%5E%7B%2A%7D%20%3D%20k%5E%7Bn%20-1%7DA%5E%7B%2A%7D%7B%5C%20%5C%20%7D%5Cleft%28%20n%20%5Cgeq%202%20%5Cright%29&id=SbXPC)

但%5E%7B%7D%20%3D%20A%5E%7B%7D%20%5Cpm%20B%5E%7B*%7D#card=math&code=%5Cleft%28%20A%20%5Cpm%20B%20%5Cright%29%5E%7B%2A%7D%20%3D%20A%5E%7B%2A%7D%20%5Cpm%20B%5E%7B%2A%7D&id=B1blF)不一定成立。

(4) %7D%5E%7BT%7D%20%3D%20%7B(A%5E%7BT%7D)%7D%5E%7B-%201%7D%2C%5C%20%5Cleft(%20A%5E%7B-%201%7D%20%5Cright)%5E%7B%7D%20%3D%7B(AA%5E%7B%7D)%7D%5E%7B-%201%7D%2C%7B(A%5E%7B%7D)%7D%5E%7BT%7D%20%3D%20%5Cleft(%20A%5E%7BT%7D%20%5Cright)%5E%7B%7D#card=math&code=%7B%28A%5E%7B-%201%7D%29%7D%5E%7BT%7D%20%3D%20%7B%28A%5E%7BT%7D%29%7D%5E%7B-%201%7D%2C%5C%20%5Cleft%28%20A%5E%7B-%201%7D%20%5Cright%29%5E%7B%2A%7D%20%3D%7B%28AA%5E%7B%2A%7D%29%7D%5E%7B-%201%7D%2C%7B%28A%5E%7B%2A%7D%29%7D%5E%7BT%7D%20%3D%20%5Cleft%28%20A%5E%7BT%7D%20%5Cright%29%5E%7B%2A%7D&id=G7anY)

5.有关的结论

(1)

(2) %2C%5C%20%5C%20%5C%20%5C%20%7B(kA)%7D%5E%7B%7D%20%3D%20k%5E%7Bn%20-1%7DA%5E%7B%7D%2C%7B%7B%5C%20%5C%20%7D%5Cleft(%20A%5E%7B%7D%20%5Cright)%7D%5E%7B%7D%20%3D%20%7CA%7C%5E%7Bn%20-%202%7DA(n%20%5Cgeq%203)#card=math&code=%7CA%5E%7B%2A%7D%7C%20%3D%20%7CA%7C%5E%7Bn%20-%201%7D%5C%20%28n%20%5Cgeq%202%29%2C%5C%20%5C%20%5C%20%5C%20%7B%28kA%29%7D%5E%7B%2A%7D%20%3D%20k%5E%7Bn%20-1%7DA%5E%7B%2A%7D%2C%7B%7B%5C%20%5C%20%7D%5Cleft%28%20A%5E%7B%2A%7D%20%5Cright%29%7D%5E%7B%2A%7D%20%3D%20%7CA%7C%5E%7Bn%20-%202%7DA%28n%20%5Cgeq%203%29&id=io4iS)

(3) 若可逆，则%7D%5E%7B*%7D%20%3D%20%5Cfrac%7B1%7D%7B%7CA%7C%7DA#card=math&code=A%5E%7B%2A%7D%20%3D%20%7CA%7CA%5E%7B-%201%7D%2C%7B%28A%5E%7B%2A%7D%29%7D%5E%7B%2A%7D%20%3D%20%5Cfrac%7B1%7D%7B%7CA%7C%7DA&id=wZTSk)

(4) 若为阶方阵，则：

%3D%5Cbegin%7Bcases%7Dn%2C%5Cquad%20r(A)%3Dn%5C%5C%201%2C%5Cquad%20r(A)%3Dn-1%5C%5C%200%2C%5Cquad%20r(A)%3Cn-1%5Cend%7Bcases%7D#card=math&code=r%28A%5E%2A%29%3D%5Cbegin%7Bcases%7Dn%2C%5Cquad%20r%28A%29%3Dn%5C%5C%201%2C%5Cquad%20r%28A%29%3Dn-1%5C%5C%200%2C%5Cquad%20r%28A%29%3Cn-1%5Cend%7Bcases%7D&id=Zfp87)

6.有关的结论

可逆%20%3D%20n%3B#card=math&code=%5CLeftrightarrow%20AB%20%3D%20E%3B%20%5CLeftrightarrow%20%7CA%7C%20%5Cneq%200%3B%20%5CLeftrightarrow%20r%28A%29%20%3D%20n%3B&id=kG4Kp)

可以表示为初等矩阵的乘积；。

7.有关矩阵秩的结论

(1) 秩#card=math&code=r%28A%29&id=Rt4Hc)=行秩=列秩；

(2) %20%5Cleq%20%5Cmin(m%2Cn)%3B#card=math&code=r%28A_%7Bm%20%5Ctimes%20n%7D%29%20%5Cleq%20%5Cmin%28m%2Cn%29%3B&id=AadP7)

(3) %20%5Cgeq%201#card=math&code=A%20%5Cneq%200%20%5CRightarrow%20r%28A%29%20%5Cgeq%201&id=n0V75)；

(4) %20%5Cleq%20r(A)%20%2B%20r(B)%3B#card=math&code=r%28A%20%5Cpm%20B%29%20%5Cleq%20r%28A%29%20%2B%20r%28B%29%3B&id=yzNgS)

(5) 初等变换不改变矩阵的秩

(6) %20%2B%20r(B)%20-%20n%20%5Cleq%20r(AB)%20%5Cleq%20%5Cmin(r(A)%2Cr(B))%2C#card=math&code=r%28A%29%20%2B%20r%28B%29%20-%20n%20%5Cleq%20r%28AB%29%20%5Cleq%20%5Cmin%28r%28A%29%2Cr%28B%29%29%2C&id=vc8vG)特别若
则：%20%2B%20r(B)%20%5Cleq%20n#card=math&code=r%28A%29%20%2B%20r%28B%29%20%5Cleq%20n&id=oczVv)

(7) 若存在%20%3D%20r(B)%3B#card=math&code=%5CRightarrow%20r%28AB%29%20%3D%20r%28B%29%3B&id=rUwiR) 若存在
%20%3D%20r(A)%3B#card=math&code=%5CRightarrow%20r%28AB%29%20%3D%20r%28A%29%3B&id=Eglc9)

若%20%3D%20n%20%5CRightarrow%20r(AB)%20%3D%20r(B)%3B#card=math&code=r%28A%7Bm%20%5Ctimes%20n%7D%29%20%3D%20n%20%5CRightarrow%20r%28AB%29%20%3D%20r%28B%29%3B&id=Motsw) 若%20%3D%20n%5CRightarrow%20r(AB)%20%3D%20r%5Cleft(%20A%20%5Cright)#card=math&code=r%28A_%7Bm%20%5Ctimes%20s%7D%29%20%3D%20n%5CRightarrow%20r%28AB%29%20%3D%20r%5Cleft%28%20A%20%5Cright%29&id=ewutF)。

(8) %20%3D%20n%20%5CLeftrightarrow%20Ax%20%3D%200#card=math&code=r%28A_%7Bm%20%5Ctimes%20s%7D%29%20%3D%20n%20%5CLeftrightarrow%20Ax%20%3D%200&id=l6C0T)只有零解

8.分块求逆公式

； ；

；

这里，均为可逆方阵。

向量

1.有关向量组的线性表示

(1)线性相关至少有一个向量可以用其余向量线性表示。

(2)线性无关，，线性相关可以由唯一线性表示。

(3) 可以由线性表示
%20%3Dr(%5Calpha%7B1%7D%2C%5Calpha%7B2%7D%2C%5Ccdots%2C%5Calpha%7Bs%7D%2C%5Cbeta)#card=math&code=%5CLeftrightarrow%20r%28%5Calpha%7B1%7D%2C%5Calpha%7B2%7D%2C%5Ccdots%2C%5Calpha%7Bs%7D%29%20%3Dr%28%5Calpha%7B1%7D%2C%5Calpha%7B2%7D%2C%5Ccdots%2C%5Calpha_%7Bs%7D%2C%5Cbeta%29&id=HEL3i) 。

2.有关向量组的线性相关性

(1)部分相关，整体相关；整体无关，部分无关.

(2) ① 个维向量
线性无关， 个维向量线性相关

。

② 个维向量线性相关。

③ 若线性无关，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。

3.有关向量组的线性表示

(1) 线性相关至少有一个向量可以用其余向量线性表示。

(2) 线性无关，，线性相关 可以由唯一线性表示。

(3) 可以由线性表示
%20%3Dr(%5Calpha%7B1%7D%2C%5Calpha%7B2%7D%2C%5Ccdots%2C%5Calpha%7Bs%7D%2C%5Cbeta)#card=math&code=%5CLeftrightarrow%20r%28%5Calpha%7B1%7D%2C%5Calpha%7B2%7D%2C%5Ccdots%2C%5Calpha%7Bs%7D%29%20%3Dr%28%5Calpha%7B1%7D%2C%5Calpha%7B2%7D%2C%5Ccdots%2C%5Calpha_%7Bs%7D%2C%5Cbeta%29&id=q5ZDH)

4.向量组的秩与矩阵的秩之间的关系

设%20%3Dr#card=math&code=r%28A_%7Bm%20%5Ctimes%20n%7D%29%20%3Dr&id=ewUhX)，则的秩#card=math&code=r%28A%29&id=h1CLh)与的行列向量组的线性相关性关系为：

(1) 若%20%3D%20r%20%3D%20m#card=math&code=r%28A_%7Bm%20%5Ctimes%20n%7D%29%20%3D%20r%20%3D%20m&id=KVULE)，则的行向量组线性无关。

(2) 若%20%3D%20r%20%3C%20m#card=math&code=r%28A_%7Bm%20%5Ctimes%20n%7D%29%20%3D%20r%20%3C%20m&id=dtlu0)，则的行向量组线性相关。

(3) 若%20%3D%20r%20%3D%20n#card=math&code=r%28A_%7Bm%20%5Ctimes%20n%7D%29%20%3D%20r%20%3D%20n&id=h9o6M)，则的列向量组线性无关。

(4) 若%20%3D%20r%20%3C%20n#card=math&code=r%28A_%7Bm%20%5Ctimes%20n%7D%29%20%3D%20r%20%3C%20n&id=PnbvA)，则的列向量组线性相关。

5.维向量空间的基变换公式及过渡矩阵

若与是向量空间的两组基，则基变换公式为：

%20%3D%20(%5Calpha%7B1%7D%2C%5Calpha%7B2%7D%2C%5Ccdots%2C%5Calpha%7Bn%7D)%5Cbegin%7Bbmatrix%7D%20%20c%7B11%7D%26%20c%7B12%7D%26%20%5Ccdots%20%26%20c%7B1n%7D%20%5C%5C%20%20c%7B21%7D%26%20c%7B22%7D%26%5Ccdots%20%26%20c%7B2n%7D%20%5C%5C%20%5Ccdots%20%26%20%5Ccdots%20%26%20%5Ccdots%20%26%20%5Ccdots%20%5C%5C%20%20c%7Bn1%7D%26%20c%7Bn2%7D%20%26%20%5Ccdots%20%26%20c%7B%7Bnn%7D%7D%20%5C%5C%5Cend%7Bbmatrix%7D%20%3D%20(%5Calpha%7B1%7D%2C%5Calpha%7B2%7D%2C%5Ccdots%2C%5Calpha%7Bn%7D)C#card=math&code=%28%5Cbeta%7B1%7D%2C%5Cbeta%7B2%7D%2C%5Ccdots%2C%5Cbeta%7Bn%7D%29%20%3D%20%28%5Calpha%7B1%7D%2C%5Calpha%7B2%7D%2C%5Ccdots%2C%5Calpha%7Bn%7D%29%5Cbegin%7Bbmatrix%7D%20%20c%7B11%7D%26%20c%7B12%7D%26%20%5Ccdots%20%26%20c%7B1n%7D%20%5C%5C%20%20c%7B21%7D%26%20c%7B22%7D%26%5Ccdots%20%26%20c%7B2n%7D%20%5C%5C%20%5Ccdots%20%26%20%5Ccdots%20%26%20%5Ccdots%20%26%20%5Ccdots%20%5C%5C%20%20c%7Bn1%7D%26%20c%7Bn2%7D%20%26%20%5Ccdots%20%26%20c%7B%7Bnn%7D%7D%20%5C%5C%5Cend%7Bbmatrix%7D%20%3D%20%28%5Calpha%7B1%7D%2C%5Calpha%7B2%7D%2C%5Ccdots%2C%5Calpha_%7Bn%7D%29C&id=nO8e4)

其中是可逆矩阵，称为由基到基的过渡矩阵。

6.坐标变换公式

若向量在基与基的坐标分别是
%7D%5E%7BT%7D#card=math&code=X%20%3D%20%7B%28x%7B1%7D%2Cx%7B2%7D%2C%5Ccdots%2Cx_%7Bn%7D%29%7D%5E%7BT%7D&id=FiXt6)，

%5E%7BT%7D#card=math&code=Y%20%3D%20%5Cleft%28%20y%7B1%7D%2Cy%7B2%7D%2C%5Ccdots%2Cy%7Bn%7D%20%5Cright%29%5E%7BT%7D&id=o1LXz) 即： ，则向量坐标变换公式为 或，其中是从基到基的过渡矩阵。

7.向量的内积

%20%3D%20a%7B1%7Db%7B1%7D%20%2B%20a%7B2%7Db%7B2%7D%20%2B%20%5Ccdots%20%2B%20a%7Bn%7Db%7Bn%7D%20%3D%20%5Calpha%5E%7BT%7D%5Cbeta%20%3D%20%5Cbeta%5E%7BT%7D%5Calpha#card=math&code=%28%5Calpha%2C%5Cbeta%29%20%3D%20a%7B1%7Db%7B1%7D%20%2B%20a%7B2%7Db%7B2%7D%20%2B%20%5Ccdots%20%2B%20a%7Bn%7Db%7Bn%7D%20%3D%20%5Calpha%5E%7BT%7D%5Cbeta%20%3D%20%5Cbeta%5E%7BT%7D%5Calpha&id=w653A)

8.Schmidt正交化

若线性无关，则可构造使其两两正交，且仅是的线性组合#card=math&code=%28i%3D%201%2C2%2C%5Ccdots%2Cn%29&id=gSwzZ)，再把单位化，记，则是规范正交向量组。其中
， %7D%7B(%5Cbeta%7B1%7D%2C%5Cbeta%7B1%7D)%7D%5Cbeta%7B1%7D#card=math&code=%5Cbeta%7B2%7D%20%3D%20%5Calpha%7B2%7D%20-%5Cfrac%7B%28%5Calpha%7B2%7D%2C%5Cbeta%7B1%7D%29%7D%7B%28%5Cbeta%7B1%7D%2C%5Cbeta%7B1%7D%29%7D%5Cbeta%7B1%7D&id=YxUSw) ， %7D%7B(%5Cbeta%7B1%7D%2C%5Cbeta%7B1%7D)%7D%5Cbeta%7B1%7D%20-%5Cfrac%7B(%5Calpha%7B3%7D%2C%5Cbeta%7B2%7D)%7D%7B(%5Cbeta%7B2%7D%2C%5Cbeta%7B2%7D)%7D%5Cbeta%7B2%7D#card=math&code=%5Cbeta%7B3%7D%20%3D%5Calpha%7B3%7D%20-%20%5Cfrac%7B%28%5Calpha%7B3%7D%2C%5Cbeta%7B1%7D%29%7D%7B%28%5Cbeta%7B1%7D%2C%5Cbeta%7B1%7D%29%7D%5Cbeta%7B1%7D%20-%5Cfrac%7B%28%5Calpha%7B3%7D%2C%5Cbeta%7B2%7D%29%7D%7B%28%5Cbeta%7B2%7D%2C%5Cbeta%7B2%7D%29%7D%5Cbeta%7B2%7D&id=qUsvS) ，

…………

%7D%7B(%5Cbeta%7B1%7D%2C%5Cbeta%7B1%7D)%7D%5Cbeta%7B1%7D%20-%20%5Cfrac%7B(%5Calpha%7Bs%7D%2C%5Cbeta%7B2%7D)%7D%7B(%5Cbeta%7B2%7D%2C%5Cbeta%7B2%7D)%7D%5Cbeta%7B2%7D%20-%20%5Ccdots%20-%20%5Cfrac%7B(%5Calpha%7Bs%7D%2C%5Cbeta%7Bs%20-%201%7D)%7D%7B(%5Cbeta%7Bs%20-%201%7D%2C%5Cbeta%7Bs%20-%201%7D)%7D%5Cbeta%7Bs%20-%201%7D#card=math&code=%5Cbeta%7Bs%7D%20%3D%20%5Calpha%7Bs%7D%20-%20%5Cfrac%7B%28%5Calpha%7Bs%7D%2C%5Cbeta%7B1%7D%29%7D%7B%28%5Cbeta%7B1%7D%2C%5Cbeta%7B1%7D%29%7D%5Cbeta%7B1%7D%20-%20%5Cfrac%7B%28%5Calpha%7Bs%7D%2C%5Cbeta%7B2%7D%29%7D%7B%28%5Cbeta%7B2%7D%2C%5Cbeta%7B2%7D%29%7D%5Cbeta%7B2%7D%20-%20%5Ccdots%20-%20%5Cfrac%7B%28%5Calpha%7Bs%7D%2C%5Cbeta%7Bs%20-%201%7D%29%7D%7B%28%5Cbeta%7Bs%20-%201%7D%2C%5Cbeta%7Bs%20-%201%7D%29%7D%5Cbeta%7Bs%20-%201%7D&id=HZsZX)

9.正交基及规范正交基

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。

线性方程组

1．克莱姆法则

线性方程组，如果系数行列式，则方程组有唯一解，，其中是把中第列元素换成方程组右端的常数列所得的行列式。

2. 阶矩阵可逆只有零解。总有唯一解，一般地，%20%3D%20n%20%5CLeftrightarrow%20Ax%3D%200#card=math&code=r%28A_%7Bm%20%5Ctimes%20n%7D%29%20%3D%20n%20%5CLeftrightarrow%20Ax%3D%200&id=dHyCd)只有零解。

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

(1) 设为矩阵，若%20%3D%20m#card=math&code=r%28A_%7Bm%20%5Ctimes%20n%7D%29%20%3D%20m&id=VO8Ql)，则对而言必有%20%3D%20r(A%20%5Cvdots%20b)%20%3D%20m#card=math&code=r%28A%29%20%3D%20r%28A%20%5Cvdots%20b%29%20%3D%20m&id=ytvFR)，从而有解。

(2) 设为的解，则当时仍为的解；但当时，则为的解。特别为的解；#card=math&code=2x%7B3%7D%20-%20%28x%7B1%7D%20%2Bx_%7B2%7D%29&id=cDPFs)为的解。

(3) 非齐次线性方程组无解%20%2B%201%20%3Dr(%5Coverline%7BA%7D)%20%5CLeftrightarrow%20b#card=math&code=%5CLeftrightarrow%20r%28A%29%20%2B%201%20%3Dr%28%5Coverline%7BA%7D%29%20%5CLeftrightarrow%20b&id=xIRyM)不能由的列向量线性表示。

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

(1) 齐次方程组恒有解(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#card=math&code=n%20-%20r%28A%29&id=jODPe)，解空间的一组基称为齐次方程组的基础解系。

(2) 是的基础解系，即：

1. 是的解；
2. 线性无关；
3. 的任一解都可以由线性表出.
   是的通解，其中是任意常数。

矩阵的特征值和特征向量

1.矩阵的特征值和特征向量的概念及性质

(1) 设是的一个特征值，则 %2CA%5E%7BT%7D%2CA%5E%7B-%201%7D%2CA%5E%7B*%7D#card=math&code=%7BkA%7D%2C%7BaA%7D%20%2B%20%7BbE%7D%2CA%5E%7B2%7D%2CA%5E%7Bm%7D%2Cf%28A%29%2CA%5E%7BT%7D%2CA%5E%7B-%201%7D%2CA%5E%7B%2A%7D&id=Q09IP)有一个特征值分别为
%2C%5Clambda%2C%5Clambda%5E%7B-%201%7D%2C%5Cfrac%7B%7CA%7C%7D%7B%5Clambda%7D%2C#card=math&code=%7Bk%CE%BB%7D%2C%7Ba%CE%BB%7D%20%2B%20b%2C%5Clambda%5E%7B2%7D%2C%5Clambda%5E%7Bm%7D%2Cf%28%5Clambda%29%2C%5Clambda%2C%5Clambda%5E%7B-%201%7D%2C%5Cfrac%7B%7CA%7C%7D%7B%5Clambda%7D%2C&id=vq7Vz)且对应特征向量相同（ 例外）。

(2)若为的个特征值，则 ,从而没有特征值。

(3)设为的个特征值，对应特征向量为，

若: ,

则: 。

2.相似变换、相似矩阵的概念及性质

(1) 若，则

1. 
2. %20%3D%20r(B)#card=math&code=%7CA%7C%20%3D%20%7CB%7C%2C%5Csum%7Bi%20%3D%201%7D%5E%7Bn%7DA%7B%7Bii%7D%7D%20%3D%20%5Csum%7Bi%20%3D1%7D%5E%7Bn%7Db%7B%7Bii%7D%7D%2Cr%28A%29%20%3D%20r%28B%29&id=EqxVB)
3. ，对成立

3.矩阵可相似对角化的充分必要条件

(1)设为阶方阵，则可对角化对每个重根特征值，有%20%3D%20k%7Bi%7D#card=math&code=n-r%28%5Clambda%7Bi%7DE%20-%20A%29%20%3D%20k_%7Bi%7D&id=L8Pkq)

(2) 设可对角化，则由有，从而

(3) 重要结论

1. 若，则.
2. 若，则%20%5Csim%20f(B)%2C%5Cleft%7C%20f(A)%20%5Cright%7C%20%5Csim%20%5Cleft%7C%20f(B)%5Cright%7C#card=math&code=f%28A%29%20%5Csim%20f%28B%29%2C%5Cleft%7C%20f%28A%29%20%5Cright%7C%20%5Csim%20%5Cleft%7C%20f%28B%29%5Cright%7C&id=cOGwC)，其中#card=math&code=f%28A%29&id=B9wsF)为关于阶方阵的多项式。
3. 若为可对角化矩阵，则其非零特征值的个数(重根重复计算)＝秩()

4.实对称矩阵的特征值、特征向量及相似对角阵

(1)相似矩阵：设为两个阶方阵，如果存在一个可逆矩阵，使得成立，则称矩阵与相似，记为。

(2)相似矩阵的性质：如果则有：

1. 
2. （若，均可逆）
3. （为正整数）
4. ，从而
   有相同的特征值
5. ，从而同时可逆或者不可逆
6. 秩%20%3D#card=math&code=%5Cleft%28%20A%20%5Cright%29%20%3D&id=kwZnL)秩%2C%5Cleft%7C%20%7B%CE%BBE%7D%20-%20A%20%5Cright%7C%20%3D%5Cleft%7C%20%7B%CE%BBE%7D%20-%20B%20%5Cright%7C#card=math&code=%5Cleft%28%20B%20%5Cright%29%2C%5Cleft%7C%20%7B%CE%BBE%7D%20-%20A%20%5Cright%7C%20%3D%5Cleft%7C%20%7B%CE%BBE%7D%20-%20B%20%5Cright%7C&id=bjzor)，不一定相似

二次型

1.个变量的二次齐次函数

%20%3D%20%5Csum%7Bi%20%3D%201%7D%5E%7Bn%7D%7B%5Csum%7Bj%20%3D1%7D%5E%7Bn%7D%7Ba%7B%7Bij%7D%7Dx%7Bi%7Dy%7Bj%7D%7D%7D#card=math&code=f%28x%7B1%7D%2Cx%7B2%7D%2C%5Ccdots%2Cx%7Bn%7D%29%20%3D%20%5Csum%7Bi%20%3D%201%7D%5E%7Bn%7D%7B%5Csum%7Bj%20%3D1%7D%5E%7Bn%7D%7Ba%7B%7Bij%7D%7Dx%7Bi%7Dy%7Bj%7D%7D%7D&id=FeHO7)，其中#card=math&code=a%7B%7Bij%7D%7D%20%3D%20a%7B%7Bji%7D%7D%28i%2Cj%20%3D1%2C2%2C%5Ccdots%2Cn%29&id=C6MDM)，称为元二次型，简称二次型. 若令,这二次型可改写成矩阵向量形式。其中称为二次型矩阵，因为#card=math&code=a%7B%7Bij%7D%7D%20%3Da_%7B%7Bji%7D%7D%28i%2Cj%20%3D1%2C2%2C%5Ccdots%2Cn%29&id=CMpyz)，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，并把矩阵的秩称为二次型的秩。

2.惯性定理，二次型的标准形和规范形

(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%20%3Dx%5E%7BT%7D%7BAx%7D#card=math&code=f%20%3D%20%5Cleft%28%20x%7B1%7D%2Cx%7B2%7D%2C%5Ccdots%2Cx_%7Bn%7D%20%5Cright%29%20%3Dx%5E%7BT%7D%7BAx%7D&id=ibEnA)经过合同变换化为

称为 #card=math&code=f%28r%20%5Cleq%20n%29&id=Y6CDc)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#card=math&code=r%28A%29&id=t2eXX)唯一确定。

(3) 规范形

任一实二次型都可经过合同变换化为规范形，其中为的秩，为正惯性指数，为负惯性指数，且规范型唯一。

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

设正定%2CA%5E%7BT%7D%2CA%5E%7B-%201%7D%2CA%5E%7B*%7D#card=math&code=%5CRightarrow%20%7BkA%7D%28k%20%3E%200%29%2CA%5E%7BT%7D%2CA%5E%7B-%201%7D%2CA%5E%7B%2A%7D&id=tddlv)正定；,可逆；，且

，正定正定，但，不一定正定

正定%20%3D%20x%5E%7BT%7D%7BAx%7D%20%3E%200%2C%5Cforall%20x%20%5Cneq%200#card=math&code=%5CLeftrightarrow%20f%28x%29%20%3D%20x%5E%7BT%7D%7BAx%7D%20%3E%200%2C%5Cforall%20x%20%5Cneq%200&id=YZP9c)

的各阶顺序主子式全大于零

的所有特征值大于零

的正惯性指数为

存在可逆阵使

存在正交矩阵，使

其中正定%2CA%5E%7BT%7D%2CA%5E%7B-%201%7D%2CA%5E%7B*%7D#card=math&code=%5CRightarrow%20%7BkA%7D%28k%20%3E0%29%2CA%5E%7BT%7D%2CA%5E%7B-%201%7D%2CA%5E%7B%2A%7D&id=S58tT)正定； 可逆；，且 。

概率论和数理统计

随机事件和概率

1.事件的关系与运算

(1) 子事件：，若发生，则发生。

(2) 相等事件：，即，且 。

(3) 和事件：（或），与中至少有一个发生。

(4) 差事件：，发生但不发生。

(5) 积事件：（或），与同时发生。

(6) 互斥事件（互不相容）：=。

(7) 互逆事件（对立事件）：

2.运算律
(1) 交换律：
(2) 结合律：%5Cbigcup%20C%3DA%5Cbigcup%20(B%5Cbigcup%20C)#card=math&code=%28A%5Cbigcup%20B%29%5Cbigcup%20C%3DA%5Cbigcup%20%28B%5Cbigcup%20C%29&id=ZZOdw)
(3) 分配律：%5Cbigcap%20C%3DA%5Cbigcap%20(B%5Cbigcap%20C)#card=math&code=%28A%5Cbigcap%20B%29%5Cbigcap%20C%3DA%5Cbigcap%20%28B%5Cbigcap%20C%29&id=lqqif)
3.德$\centerdot $摩根律

4.完全事件组

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 $

5.概率的基本公式
(1)条件概率:
%3D%5Cfrac%7BP(AB)%7D%7BP(A)%7D#card=math&code=P%28B%7CA%29%3D%5Cfrac%7BP%28AB%29%7D%7BP%28A%29%7D&id=FnLPp),表示发生的条件下，发生的概率。
(2)全概率公式：
$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 $
(3) Bayes公式：

%3D%5Cfrac%7BP(A%7C%7B%7BB%7D%7Bj%7D%7D)P(%7B%7BB%7D%7Bj%7D%7D)%7D%7B%5Csum%5Climits%7Bi%3D1%7D%5E%7Bn%7D%7BP(A%7C%7B%7BB%7D%7Bi%7D%7D)P(%7B%7BB%7D%7Bi%7D%7D)%7D%7D%2Cj%3D1%2C2%2C%5Ccdots%20%2Cn#card=math&code=P%28%7B%7BB%7D%7Bj%7D%7D%7CA%29%3D%5Cfrac%7BP%28A%7C%7B%7BB%7D%7Bj%7D%7D%29P%28%7B%7BB%7D%7Bj%7D%7D%29%7D%7B%5Csum%5Climits%7Bi%3D1%7D%5E%7Bn%7D%7BP%28A%7C%7B%7BB%7D%7Bi%7D%7D%29P%28%7B%7BB%7D%7Bi%7D%7D%29%7D%7D%2Cj%3D1%2C2%2C%5Ccdots%20%2Cn&id=CQaCE)
注：上述公式中事件的个数可为可列个。
(4)乘法公式：
%3DP(%7B%7BA%7D%7B1%7D%7D)P(%7B%7BA%7D%7B2%7D%7D%7C%7B%7BA%7D%7B1%7D%7D)%3DP(%7B%7BA%7D%7B2%7D%7D)P(%7B%7BA%7D%7B1%7D%7D%7C%7B%7BA%7D%7B2%7D%7D)#card=math&code=P%28%7B%7BA%7D%7B1%7D%7D%7B%7BA%7D%7B2%7D%7D%29%3DP%28%7B%7BA%7D%7B1%7D%7D%29P%28%7B%7BA%7D%7B2%7D%7D%7C%7B%7BA%7D%7B1%7D%7D%29%3DP%28%7B%7BA%7D%7B2%7D%7D%29P%28%7B%7BA%7D%7B1%7D%7D%7C%7B%7BA%7D%7B2%7D%7D%29&id=av0wq)
%3DP(%7B%7BA%7D%7B1%7D%7D)P(%7B%7BA%7D%7B2%7D%7D%7C%7B%7BA%7D%7B1%7D%7D)P(%7B%7BA%7D%7B3%7D%7D%7C%7B%7BA%7D%7B1%7D%7D%7B%7BA%7D%7B2%7D%7D)%5Ccdots%20P(%7B%7BA%7D%7Bn%7D%7D%7C%7B%7BA%7D%7B1%7D%7D%7B%7BA%7D%7B2%7D%7D%5Ccdots%20%7B%7BA%7D%7Bn-1%7D%7D)#card=math&code=P%28%7B%7BA%7D%7B1%7D%7D%7B%7BA%7D%7B2%7D%7D%5Ccdots%20%7B%7BA%7D%7Bn%7D%7D%29%3DP%28%7B%7BA%7D%7B1%7D%7D%29P%28%7B%7BA%7D%7B2%7D%7D%7C%7B%7BA%7D%7B1%7D%7D%29P%28%7B%7BA%7D%7B3%7D%7D%7C%7B%7BA%7D%7B1%7D%7D%7B%7BA%7D%7B2%7D%7D%29%5Ccdots%20P%28%7B%7BA%7D%7Bn%7D%7D%7C%7B%7BA%7D%7B1%7D%7D%7B%7BA%7D%7B2%7D%7D%5Ccdots%20%7B%7BA%7D_%7Bn-1%7D%7D%29&id=CbSfW)

6.事件的独立性
(1)与相互独立%3DP(A)P(B)#card=math&code=%5CLeftrightarrow%20P%28AB%29%3DP%28A%29P%28B%29&id=uU49C)
(2)，，两两独立
%3DP(A)P(B)#card=math&code=%5CLeftrightarrow%20P%28AB%29%3DP%28A%29P%28B%29&id=eqbpc);%3DP(B)P(C)#card=math&code=P%28BC%29%3DP%28B%29P%28C%29&id=r964h) ;%3DP(A)P(C)#card=math&code=P%28AC%29%3DP%28A%29P%28C%29&id=aVKAT);
(3)，，相互独立
%3DP(A)P(B)#card=math&code=%5CLeftrightarrow%20P%28AB%29%3DP%28A%29P%28B%29&id=bsX0V); %3DP(B)P(C)#card=math&code=P%28BC%29%3DP%28B%29P%28C%29&id=lPV1t) ;
%3DP(A)P(C)#card=math&code=P%28AC%29%3DP%28A%29P%28C%29&id=QJraY) ; %3DP(A)P(B)P(C)#card=math&code=P%28ABC%29%3DP%28A%29P%28B%29P%28C%29&id=DZ8BJ)

7.独立重复试验

将某试验独立重复次，若每次实验中事件A发生的概率为，则次试验中发生次的概率为：
%3DC%7Bn%7D%5E%7Bk%7D%7B%7Bp%7D%5E%7Bk%7D%7D%7B%7B(1-p)%7D%5E%7Bn-k%7D%7D#card=math&code=P%28X%3Dk%29%3DC%7Bn%7D%5E%7Bk%7D%7B%7Bp%7D%5E%7Bk%7D%7D%7B%7B%281-p%29%7D%5E%7Bn-k%7D%7D&id=HPA1q)
8.重要公式与结论
P(%5Cbar%7BA%7D)%3D1-P(A)#card=math&code=%281%29P%28%5Cbar%7BA%7D%29%3D1-P%28A%29&id=zWiNt)
P(A%5Cbigcup%20B)%3DP(A)%2BP(B)-P(AB)#card=math&code=%282%29P%28A%5Cbigcup%20B%29%3DP%28A%29%2BP%28B%29-P%28AB%29&id=jS7ZF)
%3DP(A)%2BP(B)%2BP(C)-P(AB)-P(BC)-P(AC)%2BP(ABC)#card=math&code=P%28A%5Cbigcup%20B%5Cbigcup%20C%29%3DP%28A%29%2BP%28B%29%2BP%28C%29-P%28AB%29-P%28BC%29-P%28AC%29%2BP%28ABC%29&id=mFBWw)
P(A-B)%3DP(A)-P(AB)#card=math&code=%283%29P%28A-B%29%3DP%28A%29-P%28AB%29&id=et8X7)
P(A%5Cbar%7BB%7D)%3DP(A)-P(AB)%2CP(A)%3DP(AB)%2BP(A%5Cbar%7BB%7D)%2C#card=math&code=%284%29P%28A%5Cbar%7BB%7D%29%3DP%28A%29-P%28AB%29%2CP%28A%29%3DP%28AB%29%2BP%28A%5Cbar%7BB%7D%29%2C&id=eZYyM)
%3DP(A)%2BP(%5Cbar%7BA%7DB)%3DP(AB)%2BP(A%5Cbar%7BB%7D)%2BP(%5Cbar%7BA%7DB)#card=math&code=P%28A%5Cbigcup%20B%29%3DP%28A%29%2BP%28%5Cbar%7BA%7DB%29%3DP%28AB%29%2BP%28A%5Cbar%7BB%7D%29%2BP%28%5Cbar%7BA%7DB%29&id=mkSMn)
(5)条件概率#card=math&code=P%28%5Ccenterdot%20%7CB%29&id=PjAvz)满足概率的所有性质，
例如：. %3D1-P(%7B%7BA%7D%7B1%7D%7D%7CB)#card=math&code=P%28%7B%7B%5Cbar%7BA%7D%7D%7B1%7D%7D%7CB%29%3D1-P%28%7B%7BA%7D%7B1%7D%7D%7CB%29&id=Wwfpr)
%3DP(%7B%7BA%7D%7B1%7D%7D%7CB)%2BP(%7B%7BA%7D%7B2%7D%7D%7CB)-P(%7B%7BA%7D%7B1%7D%7D%7B%7BA%7D%7B2%7D%7D%7CB)#card=math&code=P%28%7B%7BA%7D%7B1%7D%7D%5Cbigcup%20%7B%7BA%7D%7B2%7D%7D%7CB%29%3DP%28%7B%7BA%7D%7B1%7D%7D%7CB%29%2BP%28%7B%7BA%7D%7B2%7D%7D%7CB%29-P%28%7B%7BA%7D%7B1%7D%7D%7B%7BA%7D%7B2%7D%7D%7CB%29&id=qTUlf)
%3DP(%7B%7BA%7D%7B1%7D%7D%7CB)P(%7B%7BA%7D%7B2%7D%7D%7C%7B%7BA%7D%7B1%7D%7DB)#card=math&code=P%28%7B%7BA%7D%7B1%7D%7D%7B%7BA%7D%7B2%7D%7D%7CB%29%3DP%28%7B%7BA%7D%7B1%7D%7D%7CB%29P%28%7B%7BA%7D%7B2%7D%7D%7C%7B%7BA%7D%7B1%7D%7DB%29&id=Vm8OY)
(6)若相互独立，则%3D%5Cprod%5Climits%7Bi%3D1%7D%5E%7Bn%7D%7BP(%7B%7BA%7D%7Bi%7D%7D)%7D%2C#card=math&code=P%28%5Cbigcap%5Climits%7Bi%3D1%7D%5E%7Bn%7D%7B%7B%7BA%7D%7Bi%7D%7D%7D%29%3D%5Cprod%5Climits%7Bi%3D1%7D%5E%7Bn%7D%7BP%28%7B%7BA%7D%7Bi%7D%7D%29%7D%2C&id=aPTdI)
%3D%5Cprod%5Climits%7Bi%3D1%7D%5E%7Bn%7D%7B(1-P(%7B%7BA%7D%7Bi%7D%7D))%7D#card=math&code=P%28%5Cbigcup%5Climits%7Bi%3D1%7D%5E%7Bn%7D%7B%7B%7BA%7D%7Bi%7D%7D%7D%29%3D%5Cprod%5Climits%7Bi%3D1%7D%5E%7Bn%7D%7B%281-P%28%7B%7BA%7D%7Bi%7D%7D%29%29%7D&id=lmgrm)
(7)互斥、互逆与独立性之间的关系：
与互逆 与互斥，但反之不成立，与互斥（或互逆）且均非零概率事件$\Rightarrow $$AB%E8%8B%A5#card=math&code=%E4%B8%8D%E7%8B%AC%E7%AB%8B.%0A%288%29%E8%8B%A5&id=XmMlD){{A}{1}},{{A}{2}},\cdots ,{{A}{m}},{{B}{1}},{{B}{2}},\cdots ,{{B}{n}}f({{A}{1}},{{A}{2}},\cdots ,{{A}{m}})_g({{B}{1}},{{B}{2}},\cdots ,{{B}{n}})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.

随机变量及其概率分布

1.随机变量及概率分布

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

2.分布函数的概念与性质

定义： %20%3D%20P(X%20%5Cleq%20x)%2C%20-%20%5Cinfty%20%3C%20x%20%3C%20%2B%20%5Cinfty#card=math&code=F%28x%29%20%3D%20P%28X%20%5Cleq%20x%29%2C%20-%20%5Cinfty%20%3C%20x%20%3C%20%2B%20%5Cinfty&id=NZLRE)

性质：(1)%20%5Cleq%201#card=math&code=0%20%5Cleq%20F%28x%29%20%5Cleq%201&id=b0mQE)

(2) #card=math&code=F%28x%29&id=G5tWM)单调不减

(3) 右连续%20%3D%20F(x)#card=math&code=F%28x%20%2B%200%29%20%3D%20F%28x%29&id=IJmq0)

(4) %20%3D%200%2CF(%20%2B%20%5Cinfty)%20%3D%201#card=math&code=F%28%20-%20%5Cinfty%29%20%3D%200%2CF%28%20%2B%20%5Cinfty%29%20%3D%201&id=ycd15)

3.离散型随机变量的概率分布

%20%3D%20p%7Bi%7D%2Ci%20%3D%201%2C2%2C%5Ccdots%2Cn%2C%5Ccdots%5Cquad%5Cquad%20p%7Bi%7D%20%5Cgeq%200%2C%5Csum%7Bi%20%3D1%7D%5E%7B%5Cinfty%7Dp%7Bi%7D%20%3D%201#card=math&code=P%28X%20%3D%20x%7Bi%7D%29%20%3D%20p%7Bi%7D%2Ci%20%3D%201%2C2%2C%5Ccdots%2Cn%2C%5Ccdots%5Cquad%5Cquad%20p%7Bi%7D%20%5Cgeq%200%2C%5Csum%7Bi%20%3D1%7D%5E%7B%5Cinfty%7Dp_%7Bi%7D%20%3D%201&id=UMF67)

4.连续型随机变量的概率密度

概率密度#card=math&code=f%28x%29&id=aKSng);非负可积，且:

(1)%20%5Cgeq%200%2C#card=math&code=f%28x%29%20%5Cgeq%200%2C&id=ctd5m)

(2)%7Bdx%7D%20%3D%201%7D#card=math&code=%5Cint_%7B-%20%5Cinfty%7D%5E%7B%2B%5Cinfty%7D%7Bf%28x%29%7Bdx%7D%20%3D%201%7D&id=apKZC)

(3)为#card=math&code=f%28x%29&id=bkVvf)的连续点，则:

%20%3D%20F’(x)#card=math&code=f%28x%29%20%3D%20F%27%28x%29&id=eGRSG)分布函数%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7Bx%7D%7Bf(t)%7Bdt%7D%7D#card=math&code=F%28x%29%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7Bx%7D%7Bf%28t%29%7Bdt%7D%7D&id=Rdmat)

5.常见分布

(1) 0-1分布:%20%3D%20p%5E%7Bk%7D%7B(1%20-%20p)%7D%5E%7B1%20-%20k%7D%2Ck%20%3D%200%2C1#card=math&code=P%28X%20%3D%20k%29%20%3D%20p%5E%7Bk%7D%7B%281%20-%20p%29%7D%5E%7B1%20-%20k%7D%2Ck%20%3D%200%2C1&id=oHXFb)

(2) 二项分布:#card=math&code=B%28n%2Cp%29&id=cUaC2)： %20%3D%20C%7Bn%7D%5E%7Bk%7Dp%5E%7Bk%7D%7B(1%20-%20p)%7D%5E%7Bn%20-%20k%7D%2Ck%20%3D0%2C1%2C%5Ccdots%2Cn#card=math&code=P%28X%20%3D%20k%29%20%3D%20C%7Bn%7D%5E%7Bk%7Dp%5E%7Bk%7D%7B%281%20-%20p%29%7D%5E%7Bn%20-%20k%7D%2Ck%20%3D0%2C1%2C%5Ccdots%2Cn&id=SjNYk)

(3) Poisson分布:#card=math&code=p%28%5Clambda%29&id=zvKJf)： %20%3D%20%5Cfrac%7B%5Clambda%5E%7Bk%7D%7D%7Bk!%7De%5E%7B-%5Clambda%7D%2C%5Clambda%20%3E%200%2Ck%20%3D%200%2C1%2C2%5Ccdots#card=math&code=P%28X%20%3D%20k%29%20%3D%20%5Cfrac%7B%5Clambda%5E%7Bk%7D%7D%7Bk%21%7De%5E%7B-%5Clambda%7D%2C%5Clambda%20%3E%200%2Ck%20%3D%200%2C1%2C2%5Ccdots&id=wDhVK)

(4) 均匀分布#card=math&code=U%28a%2Cb%29&id=eeigh)：$f(x) = { \begin{matrix} & \frac{1}{b - a},a < x< b \ & 0, \ \end{matrix} $

(5) 正态分布:%3A#card=math&code=N%28%5Cmu%2C%5Csigma%5E%7B2%7D%29%3A&id=nyvlN) %20%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%5Csigma%7De%5E%7B-%20%5Cfrac%7B%7B(x%20-%20%5Cmu)%7D%5E%7B2%7D%7D%7B2%5Csigma%5E%7B2%7D%7D%7D%2C%5Csigma%20%3E%200%2C%5Cinfty%20%3C%20x%20%3C%20%2B%20%5Cinfty#card=math&code=%5Cvarphi%28x%29%20%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%5Csigma%7De%5E%7B-%20%5Cfrac%7B%7B%28x%20-%20%5Cmu%29%7D%5E%7B2%7D%7D%7B2%5Csigma%5E%7B2%7D%7D%7D%2C%5Csigma%20%3E%200%2C%5Cinfty%20%3C%20x%20%3C%20%2B%20%5Cinfty&id=misaw)

(6)指数分布:$E(\lambda):f(x) ={ \begin{matrix} & \lambda e^{-{λx}},x > 0,\lambda > 0 \ & 0, \ \end{matrix} $

(7)几何分布:%3AP(X%20%3D%20k)%20%3D%20%7B(1%20-%20p)%7D%5E%7Bk%20-%201%7Dp%2C0%20%3C%20p%20%3C%201%2Ck%20%3D%201%2C2%2C%5Ccdots.#card=math&code=G%28p%29%3AP%28X%20%3D%20k%29%20%3D%20%7B%281%20-%20p%29%7D%5E%7Bk%20-%201%7Dp%2C0%20%3C%20p%20%3C%201%2Ck%20%3D%201%2C2%2C%5Ccdots.&id=ZDnjW)

(8)超几何分布: %3AP(X%20%3D%20k)%20%3D%20%5Cfrac%7BC%7BM%7D%5E%7Bk%7DC%7BN%20-%20M%7D%5E%7Bn%20-k%7D%7D%7BC%7BN%7D%5E%7Bn%7D%7D%2Ck%20%3D0%2C1%2C%5Ccdots%2Cmin(n%2CM)#card=math&code=H%28N%2CM%2Cn%29%3AP%28X%20%3D%20k%29%20%3D%20%5Cfrac%7BC%7BM%7D%5E%7Bk%7DC%7BN%20-%20M%7D%5E%7Bn%20-k%7D%7D%7BC%7BN%7D%5E%7Bn%7D%7D%2Ck%20%3D0%2C1%2C%5Ccdots%2Cmin%28n%2CM%29&id=Ko8Li)

6.随机变量函数的概率分布

(1)离散型：%20%3D%20p%7Bi%7D%2CY%20%3D%20g(X)#card=math&code=P%28X%20%3D%20x%7B1%7D%29%20%3D%20p_%7Bi%7D%2CY%20%3D%20g%28X%29&id=garsU)

则: %20%3D%20%5Csum%7Bg(x%7Bi%7D)%20%3D%20y%7Bi%7D%7D%5E%7B%7D%7BP(X%20%3D%20x%7Bi%7D)%7D#card=math&code=P%28Y%20%3D%20y%7Bj%7D%29%20%3D%20%5Csum%7Bg%28x%7Bi%7D%29%20%3D%20y%7Bi%7D%7D%5E%7B%7D%7BP%28X%20%3D%20x_%7Bi%7D%29%7D&id=u4YwM)

(2)连续型：%2CY%20%3D%20g(x)#card=math&code=X%5Ctilde%7B%5C%20%7Df_%7BX%7D%28x%29%2CY%20%3D%20g%28x%29&id=AaYVS)

则:%20%3D%20P(Y%20%5Cleq%20y)%20%3D%20P(g(X)%20%5Cleq%20y)%20%3D%20%5Cint%7Bg(x)%20%5Cleq%20y%7D%5E%7B%7D%7Bf%7Bx%7D(x)dx%7D#card=math&code=F%7By%7D%28y%29%20%3D%20P%28Y%20%5Cleq%20y%29%20%3D%20P%28g%28X%29%20%5Cleq%20y%29%20%3D%20%5Cint%7Bg%28x%29%20%5Cleq%20y%7D%5E%7B%7D%7Bf%7Bx%7D%28x%29dx%7D&id=PMTYe)， %20%3D%20F’%7BY%7D(y)#card=math&code=f%7BY%7D%28y%29%20%3D%20F%27_%7BY%7D%28y%29&id=oIR2y)

7.重要公式与结论

(1) %20%5CRightarrow%20%5Cvarphi(0)%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%7D%2C%5CPhi(0)%20%3D%5Cfrac%7B1%7D%7B2%7D%2C#card=math&code=X%5Csim%20N%280%2C1%29%20%5CRightarrow%20%5Cvarphi%280%29%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%7D%2C%5CPhi%280%29%20%3D%5Cfrac%7B1%7D%7B2%7D%2C&id=VHUow) %20%3D%20P(X%20%5Cleq%20-%20a)%20%3D%201%20-%20%5CPhi(a)#card=math&code=%5CPhi%28%20-%20a%29%20%3D%20P%28X%20%5Cleq%20-%20a%29%20%3D%201%20-%20%5CPhi%28a%29&id=G4epC)

(2) %20%5CRightarrow%20%5Cfrac%7BX%20-%5Cmu%7D%7B%5Csigma%7D%5Csim%20N%5Cleft(%200%2C1%20%5Cright)%2CP(X%20%5Cleq%20a)%20%3D%20%5CPhi(%5Cfrac%7Ba%20-%5Cmu%7D%7B%5Csigma%7D)#card=math&code=X%5Csim%20N%5Cleft%28%20%5Cmu%2C%5Csigma%5E%7B2%7D%20%5Cright%29%20%5CRightarrow%20%5Cfrac%7BX%20-%5Cmu%7D%7B%5Csigma%7D%5Csim%20N%5Cleft%28%200%2C1%20%5Cright%29%2CP%28X%20%5Cleq%20a%29%20%3D%20%5CPhi%28%5Cfrac%7Ba%20-%5Cmu%7D%7B%5Csigma%7D%29&id=Oa5AR)

(3) %20%5CRightarrow%20P(X%20%3E%20s%20%2B%20t%7CX%20%3E%20s)%20%3D%20P(X%20%3E%20t)#card=math&code=X%5Csim%20E%28%5Clambda%29%20%5CRightarrow%20P%28X%20%3E%20s%20%2B%20t%7CX%20%3E%20s%29%20%3D%20P%28X%20%3E%20t%29&id=ecT95)

(4) %20%5CRightarrow%20P(X%20%3D%20m%20%2B%20k%7CX%20%3E%20m)%20%3D%20P(X%20%3D%20k)#card=math&code=X%5Csim%20G%28p%29%20%5CRightarrow%20P%28X%20%3D%20m%20%2B%20k%7CX%20%3E%20m%29%20%3D%20P%28X%20%3D%20k%29&id=lHA4O)

(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。

多维随机变量及其分布

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#card=math&code=%28X%2CY%29&id=hQE9q)， 联合分布为%20%3D%20P(X%20%5Cleq%20x%2CY%20%5Cleq%20y)#card=math&code=F%28x%2Cy%29%20%3D%20P%28X%20%5Cleq%20x%2CY%20%5Cleq%20y%29&id=nQUKG)

2.二维离散型随机变量的分布

(1) 联合概率分布律

(2) 边缘分布律

(3) 条件分布律

3. 二维连续性随机变量的密度

(1) 联合概率密度%3A#card=math&code=f%28x%2Cy%29%3A&id=swmMR)

1. %20%5Cgeq%200#card=math&code=f%28x%2Cy%29%20%5Cgeq%200&id=g96KO)
2. dxdy%7D%7D%20%3D%201#card=math&code=%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7B%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bf%28x%2Cy%29dxdy%7D%7D%20%3D%201&id=hDvdK)

(2) 分布函数：%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7Bx%7D%7B%5Cint%7B-%20%5Cinfty%7D%5E%7By%7D%7Bf(u%2Cv)dudv%7D%7D#card=math&code=F%28x%2Cy%29%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7Bx%7D%7B%5Cint%7B-%20%5Cinfty%7D%5E%7By%7D%7Bf%28u%2Cv%29dudv%7D%7D&id=uPDOC)

(3) 边缘概率密度： %20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bf%5Cleft(%20x%2Cy%20%5Cright)%7Bdy%7D%7D#card=math&code=f%7BX%7D%5Cleft%28%20x%20%5Cright%29%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bf%5Cleft%28%20x%2Cy%20%5Cright%29%7Bdy%7D%7D&id=sPWrY) %20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bf(x%2Cy)dx%7D#card=math&code=f%7BY%7D%28y%29%20%3D%20%5Cint_%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bf%28x%2Cy%29dx%7D&id=dRlFE)

(4) 条件概率密度：%20%3D%20%5Cfrac%7Bf%5Cleft(%20x%2Cy%20%5Cright)%7D%7Bf%7BY%7D%5Cleft(%20y%20%5Cright)%7D#card=math&code=f%7BX%7CY%7D%5Cleft%28%20x%20%5Cmiddle%7C%20y%20%5Cright%29%20%3D%20%5Cfrac%7Bf%5Cleft%28%20x%2Cy%20%5Cright%29%7D%7Bf%7BY%7D%5Cleft%28%20y%20%5Cright%29%7D&id=E6jv7) %20%3D%20%5Cfrac%7Bf(x%2Cy)%7D%7Bf%7BX%7D(x)%7D#card=math&code=f%7BY%7CX%7D%28y%7Cx%29%20%3D%20%5Cfrac%7Bf%28x%2Cy%29%7D%7Bf_%7BX%7D%28x%29%7D&id=JnMc1)

4.常见二维随机变量的联合分布

(1) 二维均匀分布：%20%5Csim%20U(D)#card=math&code=%28x%2Cy%29%20%5Csim%20U%28D%29&id=aNpCW) ,%20%3D%20%5Cbegin%7Bcases%7D%20%5Cfrac%7B1%7D%7BS(D)%7D%2C(x%2Cy)%20%5Cin%20D%20%5C%5C%20%20%200%2C%E5%85%B6%E4%BB%96%20%20%5Cend%7Bcases%7D#card=math&code=f%28x%2Cy%29%20%3D%20%5Cbegin%7Bcases%7D%20%5Cfrac%7B1%7D%7BS%28D%29%7D%2C%28x%2Cy%29%20%5Cin%20D%20%5C%5C%20%20%200%2C%E5%85%B6%E4%BB%96%20%20%5Cend%7Bcases%7D&id=rup4L)

(2) 二维正态分布：%5Csim%20N(%5Cmu%7B1%7D%2C%5Cmu%7B2%7D%2C%5Csigma%7B1%7D%5E%7B2%7D%2C%5Csigma%7B2%7D%5E%7B2%7D%2C%5Crho)#card=math&code=%28X%2CY%29%5Csim%20N%28%5Cmu%7B1%7D%2C%5Cmu%7B2%7D%2C%5Csigma%7B1%7D%5E%7B2%7D%2C%5Csigma%7B2%7D%5E%7B2%7D%2C%5Crho%29&id=fEFHE),%5Csim%20N(%5Cmu%7B1%7D%2C%5Cmu%7B2%7D%2C%5Csigma%7B1%7D%5E%7B2%7D%2C%5Csigma%7B2%7D%5E%7B2%7D%2C%5Crho)#card=math&code=%28X%2CY%29%5Csim%20N%28%5Cmu%7B1%7D%2C%5Cmu%7B2%7D%2C%5Csigma%7B1%7D%5E%7B2%7D%2C%5Csigma%7B2%7D%5E%7B2%7D%2C%5Crho%29&id=T08iT)

%20%3D%20%5Cfrac%7B1%7D%7B2%5Cpi%5Csigma%7B1%7D%5Csigma%7B2%7D%5Csqrt%7B1%20-%20%5Crho%5E%7B2%7D%7D%7D.%5Cexp%5Cleft%5C%7B%20%5Cfrac%7B-%201%7D%7B2(1%20-%20%5Crho%5E%7B2%7D)%7D%5Clbrack%5Cfrac%7B%7B(x%20-%20%5Cmu%7B1%7D)%7D%5E%7B2%7D%7D%7B%5Csigma%7B1%7D%5E%7B2%7D%7D%20-%202%5Crho%5Cfrac%7B(x%20-%20%5Cmu%7B1%7D)(y%20-%20%5Cmu%7B2%7D)%7D%7B%5Csigma%7B1%7D%5Csigma%7B2%7D%7D%20%2B%20%5Cfrac%7B%7B(y%20-%20%5Cmu%7B2%7D)%7D%5E%7B2%7D%7D%7B%5Csigma%7B2%7D%5E%7B2%7D%7D%5Crbrack%20%5Cright%5C%7D#card=math&code=f%28x%2Cy%29%20%3D%20%5Cfrac%7B1%7D%7B2%5Cpi%5Csigma%7B1%7D%5Csigma%7B2%7D%5Csqrt%7B1%20-%20%5Crho%5E%7B2%7D%7D%7D.%5Cexp%5Cleft%5C%7B%20%5Cfrac%7B-%201%7D%7B2%281%20-%20%5Crho%5E%7B2%7D%29%7D%5Clbrack%5Cfrac%7B%7B%28x%20-%20%5Cmu%7B1%7D%29%7D%5E%7B2%7D%7D%7B%5Csigma%7B1%7D%5E%7B2%7D%7D%20-%202%5Crho%5Cfrac%7B%28x%20-%20%5Cmu%7B1%7D%29%28y%20-%20%5Cmu%7B2%7D%29%7D%7B%5Csigma%7B1%7D%5Csigma%7B2%7D%7D%20%2B%20%5Cfrac%7B%7B%28y%20-%20%5Cmu%7B2%7D%29%7D%5E%7B2%7D%7D%7B%5Csigma%7B2%7D%5E%7B2%7D%7D%5Crbrack%20%5Cright%5C%7D&id=pwpBW)

5.随机变量的独立性和相关性

和的相互独立:%20%3D%20F%7BX%7D%5Cleft(%20x%20%5Cright)F%7BY%7D%5Cleft(%20y%20%5Cright)#card=math&code=%5CLeftrightarrow%20F%5Cleft%28%20x%2Cy%20%5Cright%29%20%3D%20F%7BX%7D%5Cleft%28%20x%20%5Cright%29F%7BY%7D%5Cleft%28%20y%20%5Cright%29&id=JeOTi):

（离散型）
%20%3D%20f%7BX%7D%5Cleft(%20x%20%5Cright)f%7BY%7D%5Cleft(%20y%20%5Cright)#card=math&code=%5CLeftrightarrow%20f%5Cleft%28%20x%2Cy%20%5Cright%29%20%3D%20f%7BX%7D%5Cleft%28%20x%20%5Cright%29f%7BY%7D%5Cleft%28%20y%20%5Cright%29&id=tGqZU)（连续型）

和的相关性：

相关系数时，称和不相关，
否则称和相关

6.两个随机变量简单函数的概率分布

离散型： %20%3D%20p%7B%7Bij%7D%7D%2CZ%20%3D%20g%5Cleft(%20X%2CY%20%5Cright)#card=math&code=P%5Cleft%28%20X%20%3D%20x%7Bi%7D%2CY%20%3D%20y%7Bi%7D%20%5Cright%29%20%3D%20p%7B%7Bij%7D%7D%2CZ%20%3D%20g%5Cleft%28%20X%2CY%20%5Cright%29&id=Bb6tY) 则：

%20%3D%20P%5Cleft%5C%7B%20g%5Cleft(%20X%2CY%20%5Cright)%20%3D%20z%7Bk%7D%20%5Cright%5C%7D%20%3D%20%5Csum%7Bg%5Cleft(%20x%7Bi%7D%2Cy%7Bi%7D%20%5Cright)%20%3D%20z%7Bk%7D%7D%5E%7B%7D%7BP%5Cleft(%20X%20%3D%20x%7Bi%7D%2CY%20%3D%20y%7Bj%7D%20%5Cright)%7D#card=math&code=P%28Z%20%3D%20z%7Bk%7D%29%20%3D%20P%5Cleft%5C%7B%20g%5Cleft%28%20X%2CY%20%5Cright%29%20%3D%20z%7Bk%7D%20%5Cright%5C%7D%20%3D%20%5Csum%7Bg%5Cleft%28%20x%7Bi%7D%2Cy%7Bi%7D%20%5Cright%29%20%3D%20z%7Bk%7D%7D%5E%7B%7D%7BP%5Cleft%28%20X%20%3D%20x%7Bi%7D%2CY%20%3D%20y_%7Bj%7D%20%5Cright%29%7D&id=vMK7B)

连续型： %20%5Csim%20f%5Cleft(%20x%2Cy%20%5Cright)%2CZ%20%3D%20g%5Cleft(%20X%2CY%20%5Cright)#card=math&code=%5Cleft%28%20X%2CY%20%5Cright%29%20%5Csim%20f%5Cleft%28%20x%2Cy%20%5Cright%29%2CZ%20%3D%20g%5Cleft%28%20X%2CY%20%5Cright%29&id=DaCkh)
则：

%20%3D%20P%5Cleft%5C%7B%20g%5Cleft(%20X%2CY%20%5Cright)%20%5Cleq%20z%20%5Cright%5C%7D%20%3D%20%5Ciint%7Bg(x%2Cy)%20%5Cleq%20z%7D%5E%7B%7D%7Bf(x%2Cy)dxdy%7D#card=math&code=F%7Bz%7D%5Cleft%28%20z%20%5Cright%29%20%3D%20P%5Cleft%5C%7B%20g%5Cleft%28%20X%2CY%20%5Cright%29%20%5Cleq%20z%20%5Cright%5C%7D%20%3D%20%5Ciint%7Bg%28x%2Cy%29%20%5Cleq%20z%7D%5E%7B%7D%7Bf%28x%2Cy%29dxdy%7D&id=TCv3e)，%20%3D%20F’%7Bz%7D(z)#card=math&code=f%7Bz%7D%28z%29%20%3D%20F%27_%7Bz%7D%28z%29&id=idrXP)

7.重要公式与结论

(1) 边缘密度公式： %20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bf(x%2Cy)dy%2C%7D#card=math&code=f%7BX%7D%28x%29%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bf%28x%2Cy%29dy%2C%7D&id=RPTHV)
%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bf(x%2Cy)dx%7D#card=math&code=f%7BY%7D%28y%29%20%3D%20%5Cint_%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bf%28x%2Cy%29dx%7D&id=CxesY)

(2) %20%5Cin%20D%20%5Cright%5C%7D%20%3D%20%5Ciint%7BD%7D%5E%7B%7D%7Bf%5Cleft(%20x%2Cy%20%5Cright)%7Bdxdy%7D%7D#card=math&code=P%5Cleft%5C%7B%20%5Cleft%28%20X%2CY%20%5Cright%29%20%5Cin%20D%20%5Cright%5C%7D%20%3D%20%5Ciint%7BD%7D%5E%7B%7D%7Bf%5Cleft%28%20x%2Cy%20%5Cright%29%7Bdxdy%7D%7D&id=HFMZh)

(3) 若#card=math&code=%28X%2CY%29&id=Hxdv5)服从二维正态分布#card=math&code=N%28%5Cmu%7B1%7D%2C%5Cmu%7B2%7D%2C%5Csigma%7B1%7D%5E%7B2%7D%2C%5Csigma%7B2%7D%5E%7B2%7D%2C%5Crho%29&id=LyiW6)
则有：

1. %2CY%5Csim%20N(%5Cmu%7B2%7D%2C%5Csigma%7B2%7D%5E%7B2%7D).#card=math&code=X%5Csim%20N%5Cleft%28%20%5Cmu%7B1%7D%2C%5Csigma%7B1%7D%5E%7B2%7D%20%5Cright%29%2CY%5Csim%20N%28%5Cmu%7B2%7D%2C%5Csigma%7B2%7D%5E%7B2%7D%29.&id=nFz9v)
2. 与相互独立，即与不相关。
3. #card=math&code=C%7B1%7DX%20%2B%20C%7B2%7DY%5Csim%20N%28C%7B1%7D%5Cmu%7B1%7D%20%2B%20C%7B2%7D%5Cmu%7B2%7D%2CC%7B1%7D%5E%7B2%7D%5Csigma%7B1%7D%5E%7B2%7D%20%2B%20C%7B2%7D%5E%7B2%7D%5Csigma%7B2%7D%5E%7B2%7D%20%2B%202C%7B1%7DC%7B2%7D%5Csigma%7B1%7D%5Csigma%7B2%7D%5Crho%29&id=mNQ1O)
4. 关于的条件分布为： %2C%5Csigma%7B1%7D%5E%7B2%7D(1%20-%20%5Crho%5E%7B2%7D))#card=math&code=N%28%5Cmu%7B1%7D%20%2B%20%5Crho%5Cfrac%7B%5Csigma%7B1%7D%7D%7B%5Csigma%7B2%7D%7D%28y%20-%20%5Cmu%7B2%7D%29%2C%5Csigma%7B1%7D%5E%7B2%7D%281%20-%20%5Crho%5E%7B2%7D%29%29&id=S18Ta)
5. 关于的条件分布为： %2C%5Csigma%7B2%7D%5E%7B2%7D(1%20-%20%5Crho%5E%7B2%7D))#card=math&code=N%28%5Cmu%7B2%7D%20%2B%20%5Crho%5Cfrac%7B%5Csigma%7B2%7D%7D%7B%5Csigma%7B1%7D%7D%28x%20-%20%5Cmu%7B1%7D%29%2C%5Csigma%7B2%7D%5E%7B2%7D%281%20-%20%5Crho%5E%7B2%7D%29%29&id=W726b)

(4) 若与独立，且分别服从%2CN(%5Cmu%7B1%7D%2C%5Csigma%7B2%7D%5E%7B2%7D)%2C#card=math&code=N%28%5Cmu%7B1%7D%2C%5Csigma%7B1%7D%5E%7B2%7D%29%2CN%28%5Cmu%7B1%7D%2C%5Csigma%7B2%7D%5E%7B2%7D%29%2C&id=fi9sI)
则：%5Csim%20N(%5Cmu%7B1%7D%2C%5Cmu%7B2%7D%2C%5Csigma%7B1%7D%5E%7B2%7D%2C%5Csigma%7B2%7D%5E%7B2%7D%2C0)%2C#card=math&code=%5Cleft%28%20X%2CY%20%5Cright%29%5Csim%20N%28%5Cmu%7B1%7D%2C%5Cmu%7B2%7D%2C%5Csigma%7B1%7D%5E%7B2%7D%2C%5Csigma%7B2%7D%5E%7B2%7D%2C0%29%2C&id=tcio9)

.#card=math&code=C%7B1%7DX%20%2B%20C%7B2%7DY%5Ctilde%7B%5C%20%7DN%28C%7B1%7D%5Cmu%7B1%7D%20%2B%20C%7B2%7D%5Cmu%7B2%7D%2CC%7B1%7D%5E%7B2%7D%5Csigma%7B1%7D%5E%7B2%7D%20C%7B2%7D%5E%7B2%7D%5Csigma%7B2%7D%5E%7B2%7D%29.&id=R030y)

(5) 若与相互独立，#card=math&code=f%5Cleft%28%20x%20%5Cright%29&id=uASTA)和#card=math&code=g%5Cleft%28%20x%20%5Cright%29&id=M5nmU)为连续函数， 则#card=math&code=f%5Cleft%28%20X%20%5Cright%29&id=twAIE)和#card=math&code=g%28Y%29&id=l4lkY)也相互独立。

随机变量的数字特征

1.数学期望

离散型：%20%3D%20%5Csum%7Bi%7D%5E%7B%7D%7Bx%7Bi%7Dp%7Bi%7D%7D#card=math&code=P%5Cleft%5C%7B%20X%20%3D%20x%7Bi%7D%20%5Cright%5C%7D%20%3D%20p%7Bi%7D%2CE%28X%29%20%3D%20%5Csum%7Bi%7D%5E%7B%7D%7Bx%7Bi%7Dp%7Bi%7D%7D&id=M0M9S)；

连续型： %2CE(X)%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bxf(x)dx%7D#card=math&code=X%5Csim%20f%28x%29%2CE%28X%29%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bxf%28x%29dx%7D&id=aRS3H)

性质：

(1) %20%3D%20C%2CE%5Clbrack%20E(X)%5Crbrack%20%3D%20E(X)#card=math&code=E%28C%29%20%3D%20C%2CE%5Clbrack%20E%28X%29%5Crbrack%20%3D%20E%28X%29&id=IJANQ)

(2) %20%3D%20C%7B1%7DE(X)%20%2B%20C%7B2%7DE(Y)#card=math&code=E%28C%7B1%7DX%20%2B%20C%7B2%7DY%29%20%3D%20C%7B1%7DE%28X%29%20%2B%20C%7B2%7DE%28Y%29&id=suTiJ)

(3) 若和独立，则%20%3D%20E(X)E(Y)#card=math&code=E%28XY%29%20%3D%20E%28X%29E%28Y%29&id=OFCCw)

(4)%20%5Cright%5Crbrack%5E%7B2%7D%20%5Cleq%20E(X%5E%7B2%7D)E(Y%5E%7B2%7D)#card=math&code=%5Cleft%5Clbrack%20E%28XY%29%20%5Cright%5Crbrack%5E%7B2%7D%20%5Cleq%20E%28X%5E%7B2%7D%29E%28Y%5E%7B2%7D%29&id=hmi88)

2.方差：%20%3D%20E%5Cleft%5Clbrack%20X%20-%20E(X)%20%5Cright%5Crbrack%5E%7B2%7D%20%3D%20E(X%5E%7B2%7D)%20-%20%5Cleft%5Clbrack%20E(X)%20%5Cright%5Crbrack%5E%7B2%7D#card=math&code=D%28X%29%20%3D%20E%5Cleft%5Clbrack%20X%20-%20E%28X%29%20%5Cright%5Crbrack%5E%7B2%7D%20%3D%20E%28X%5E%7B2%7D%29%20-%20%5Cleft%5Clbrack%20E%28X%29%20%5Cright%5Crbrack%5E%7B2%7D&id=VTrML)

3.标准差：%7D#card=math&code=%5Csqrt%7BD%28X%29%7D&id=WzcBv)，

4.离散型：%20%3D%20%5Csum%7Bi%7D%5E%7B%7D%7B%5Cleft%5Clbrack%20x%7Bi%7D%20-%20E(X)%20%5Cright%5Crbrack%5E%7B2%7Dp%7Bi%7D%7D#card=math&code=D%28X%29%20%3D%20%5Csum%7Bi%7D%5E%7B%7D%7B%5Cleft%5Clbrack%20x%7Bi%7D%20-%20E%28X%29%20%5Cright%5Crbrack%5E%7B2%7Dp%7Bi%7D%7D&id=vJZB8)

5.连续型：%20%3D%20%7B%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%5Cleft%5Clbrack%20x%20-%20E(X)%20%5Cright%5Crbrack%7D%5E%7B2%7Df(x)dx#card=math&code=D%28X%29%20%3D%20%7B%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%5Cleft%5Clbrack%20x%20-%20E%28X%29%20%5Cright%5Crbrack%7D%5E%7B2%7Df%28x%29dx&id=Ud53G)

性质：

(1)%20%3D%200%2CD%5Clbrack%20E(X)%5Crbrack%20%3D%200%2CD%5Clbrack%20D(X)%5Crbrack%20%3D%200#card=math&code=%5C%20D%28C%29%20%3D%200%2CD%5Clbrack%20E%28X%29%5Crbrack%20%3D%200%2CD%5Clbrack%20D%28X%29%5Crbrack%20%3D%200&id=oslfy)

(2) 与相互独立，则%20%3D%20D(X)%20%2B%20D(Y)#card=math&code=D%28X%20%5Cpm%20Y%29%20%3D%20D%28X%29%20%2B%20D%28Y%29&id=fvgxY)

(3)%20%3D%20C%7B1%7D%5E%7B2%7DD%5Cleft(%20X%20%5Cright)#card=math&code=%5C%20D%5Cleft%28%20C%7B1%7DX%20%2B%20C%7B2%7D%20%5Cright%29%20%3D%20C%7B1%7D%5E%7B2%7DD%5Cleft%28%20X%20%5Cright%29&id=tRJeu)

(4) 一般有 %20%3D%20D(X)%20%2B%20D(Y)%20%5Cpm%202Cov(X%2CY)%20%3D%20D(X)%20%2B%20D(Y)%20%5Cpm%202%5Crho%5Csqrt%7BD(X)%7D%5Csqrt%7BD(Y)%7D#card=math&code=D%28X%20%5Cpm%20Y%29%20%3D%20D%28X%29%20%2B%20D%28Y%29%20%5Cpm%202Cov%28X%2CY%29%20%3D%20D%28X%29%20%2B%20D%28Y%29%20%5Cpm%202%5Crho%5Csqrt%7BD%28X%29%7D%5Csqrt%7BD%28Y%29%7D&id=wSPkJ)

(5)%20%3C%20E%5Cleft(%20X%20-%20C%20%5Cright)%5E%7B2%7D%2CC%20%5Cneq%20E%5Cleft(%20X%20%5Cright)#card=math&code=%5C%20D%5Cleft%28%20X%20%5Cright%29%20%3C%20E%5Cleft%28%20X%20-%20C%20%5Cright%29%5E%7B2%7D%2CC%20%5Cneq%20E%5Cleft%28%20X%20%5Cright%29&id=TQpwx)

(6)%20%3D%200%20%5CLeftrightarrow%20P%5Cleft%5C%7B%20X%20%3D%20C%20%5Cright%5C%7D%20%3D%201#card=math&code=%5C%20D%28X%29%20%3D%200%20%5CLeftrightarrow%20P%5Cleft%5C%7B%20X%20%3D%20C%20%5Cright%5C%7D%20%3D%201&id=Kk1Wq)

6.随机变量函数的数学期望

(1) 对于函数#card=math&code=Y%20%3D%20g%28x%29&id=MOwDj)

为离散型：%20%3D%20%5Csum%7Bi%7D%5E%7B%7D%7Bg(x%7Bi%7D)p%7Bi%7D%7D#card=math&code=P%5C%7B%20X%20%3D%20x%7Bi%7D%5C%7D%20%3D%20p%7Bi%7D%2CE%28Y%29%20%3D%20%5Csum%7Bi%7D%5E%7B%7D%7Bg%28x%7Bi%7D%29p%7Bi%7D%7D&id=Y6mdC)；

为连续型：%2CE(Y)%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bg(x)f(x)dx%7D#card=math&code=X%5Csim%20f%28x%29%2CE%28Y%29%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bg%28x%29f%28x%29dx%7D&id=br6wI)

(2) #card=math&code=Z%20%3D%20g%28X%2CY%29&id=NUnaD);%5Csim%20P%5C%7B%20X%20%3D%20x%7Bi%7D%2CY%20%3D%20y%7Bj%7D%5C%7D%20%3D%20p%7B%7Bij%7D%7D#card=math&code=%5Cleft%28%20X%2CY%20%5Cright%29%5Csim%20P%5C%7B%20X%20%3D%20x%7Bi%7D%2CY%20%3D%20y%7Bj%7D%5C%7D%20%3D%20p%7B%7Bij%7D%7D&id=mpZqc); %20%3D%20%5Csum%7Bi%7D%5E%7B%7D%7B%5Csum%7Bj%7D%5E%7B%7D%7Bg(x%7Bi%7D%2Cy%7Bj%7D)p%7B%7Bij%7D%7D%7D%7D#card=math&code=E%28Z%29%20%3D%20%5Csum%7Bi%7D%5E%7B%7D%7B%5Csum%7Bj%7D%5E%7B%7D%7Bg%28x%7Bi%7D%2Cy%7Bj%7D%29p%7B%7Bij%7D%7D%7D%7D&id=HjW6b) %5Csim%20f(x%2Cy)#card=math&code=%5Cleft%28%20X%2CY%20%5Cright%29%5Csim%20f%28x%2Cy%29&id=buXqQ);%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7B%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bg(x%2Cy)f(x%2Cy)dxdy%7D%7D#card=math&code=E%28Z%29%20%3D%20%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7B%5Cint%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%7Bg%28x%2Cy%29f%28x%2Cy%29dxdy%7D%7D&id=w59GO)

7.协方差

%20%3D%20E%5Cleft%5Clbrack%20(X%20-%20E(X)(Y%20-%20E(Y))%20%5Cright%5Crbrack#card=math&code=Cov%28X%2CY%29%20%3D%20E%5Cleft%5Clbrack%20%28X%20-%20E%28X%29%28Y%20-%20E%28Y%29%29%20%5Cright%5Crbrack&id=JEdzZ)

8.相关系数

%7D%7B%5Csqrt%7BD(X)%7D%5Csqrt%7BD(Y)%7D%7D#card=math&code=%5Crho_%7B%7BXY%7D%7D%20%3D%20%5Cfrac%7BCov%28X%2CY%29%7D%7B%5Csqrt%7BD%28X%29%7D%5Csqrt%7BD%28Y%29%7D%7D&id=Poexd),阶原点矩 #card=math&code=E%28X%5E%7Bk%7D%29&id=M1oWn);
阶中心矩 %5Crbrack%7D%5E%7Bk%7D%20%5Cright%5C%7D#card=math&code=E%5Cleft%5C%7B%20%7B%5Clbrack%20X%20-%20E%28X%29%5Crbrack%7D%5E%7Bk%7D%20%5Cright%5C%7D&id=o8wmf)

性质：

(1)%20%3D%20Cov(Y%2CX)#card=math&code=%5C%20Cov%28X%2CY%29%20%3D%20Cov%28Y%2CX%29&id=HfsSL)

(2)%20%3D%20abCov(Y%2CX)#card=math&code=%5C%20Cov%28aX%2CbY%29%20%3D%20abCov%28Y%2CX%29&id=BIGtm)

(3)%20%3D%20Cov(X%7B1%7D%2CY)%20%2B%20Cov(X%7B2%7D%2CY)#card=math&code=%5C%20Cov%28X%7B1%7D%20%2B%20X%7B2%7D%2CY%29%20%3D%20Cov%28X%7B1%7D%2CY%29%20%2B%20Cov%28X%7B2%7D%2CY%29&id=ZdfwO)

(4)%20%5Cright%7C%20%5Cleq%201#card=math&code=%5C%20%5Cleft%7C%20%5Crho%5Cleft%28%20X%2CY%20%5Cright%29%20%5Cright%7C%20%5Cleq%201&id=okHY7)

(5) %20%3D%201%20%5CLeftrightarrow%20P%5Cleft(%20Y%20%3D%20aX%20%2B%20b%20%5Cright)%20%3D%201#card=math&code=%5C%20%5Crho%5Cleft%28%20X%2CY%20%5Cright%29%20%3D%201%20%5CLeftrightarrow%20P%5Cleft%28%20Y%20%3D%20aX%20%2B%20b%20%5Cright%29%20%3D%201&id=Sl4oO) ，其中

%20%3D%20-%201%20%5CLeftrightarrow%20P%5Cleft(%20Y%20%3D%20aX%20%2B%20b%20%5Cright)%20%3D%201#card=math&code=%5Crho%5Cleft%28%20X%2CY%20%5Cright%29%20%3D%20-%201%20%5CLeftrightarrow%20P%5Cleft%28%20Y%20%3D%20aX%20%2B%20b%20%5Cright%29%20%3D%201&id=HnrSv)
，其中

9.重要公式与结论

(1)%20%3D%20E(X%5E%7B2%7D)%20-%20E%5E%7B2%7D(X)#card=math&code=%5C%20D%28X%29%20%3D%20E%28X%5E%7B2%7D%29%20-%20E%5E%7B2%7D%28X%29&id=E1hBi)

(2)%20%3D%20E(XY)%20-%20E(X)E(Y)#card=math&code=%5C%20Cov%28X%2CY%29%20%3D%20E%28XY%29%20-%20E%28X%29E%28Y%29&id=FZ2m1)

(3) %20%5Cright%7C%20%5Cleq%201%2C#card=math&code=%5Cleft%7C%20%5Crho%5Cleft%28%20X%2CY%20%5Cright%29%20%5Cright%7C%20%5Cleq%201%2C&id=EzKUH)且 %20%3D%201%20%5CLeftrightarrow%20P%5Cleft(%20Y%20%3D%20aX%20%2B%20b%20%5Cright)%20%3D%201#card=math&code=%5Crho%5Cleft%28%20X%2CY%20%5Cright%29%20%3D%201%20%5CLeftrightarrow%20P%5Cleft%28%20Y%20%3D%20aX%20%2B%20b%20%5Cright%29%20%3D%201&id=A20Ew)，其中

%20%3D%20-%201%20%5CLeftrightarrow%20P%5Cleft(%20Y%20%3D%20aX%20%2B%20b%20%5Cright)%20%3D%201#card=math&code=%5Crho%5Cleft%28%20X%2CY%20%5Cright%29%20%3D%20-%201%20%5CLeftrightarrow%20P%5Cleft%28%20Y%20%3D%20aX%20%2B%20b%20%5Cright%29%20%3D%201&id=PQiF7)，其中

(4) 下面5个条件互为充要条件：

%20%3D%200#card=math&code=%5Crho%28X%2CY%29%20%3D%200&id=eRAlk) %20%3D%200#card=math&code=%5CLeftrightarrow%20Cov%28X%2CY%29%20%3D%200&id=uBqrM) %20%3D%20E(X)E(Y)#card=math&code=%5CLeftrightarrow%20E%28X%2CY%29%20%3D%20E%28X%29E%28Y%29&id=lCGkr) %20%3D%20D(X)%20%2B%20D(Y)#card=math&code=%5CLeftrightarrow%20D%28X%20%2B%20Y%29%20%3D%20D%28X%29%20%2B%20D%28Y%29&id=pWlkx) %20%3D%20D(X)%20%2B%20D(Y)#card=math&code=%5CLeftrightarrow%20%20D%28X%20-%20Y%29%20%3D%20D%28X%29%20%2B%20D%28Y%29&id=lF0sb)

注：与独立为上述5个条件中任何一个成立的充分条件，但非必要条件。

数理统计的基本概念

1.基本概念

总体：研究对象的全体，它是一个随机变量，用表示。

个体：组成总体的每个基本元素。

简单随机样本：来自总体的个相互独立且与总体同分布的随机变量，称为容量为的简单随机样本，简称样本。

统计量：设是来自总体的一个样本，#card=math&code=g%28X%7B1%7D%2CX%7B2%7D%5Ccdots%2CX%7Bn%7D%29&id=WGaRo)）是样本的连续函数，且#card=math&code=g%28%29&id=M31Iu)中不含任何未知参数，则称#card=math&code=g%28X%7B1%7D%2CX%7B2%7D%5Ccdots%2CX_%7Bn%7D%29&id=fjAq4)为统计量。

样本均值：

样本方差：%7D%5E%7B2%7D#card=math&code=S%5E%7B2%7D%20%3D%20%5Cfrac%7B1%7D%7Bn%20-%201%7D%5Csum%7Bi%20%3D%201%7D%5E%7Bn%7D%7B%28X%7Bi%7D%20-%20%5Coverline%7BX%7D%29%7D%5E%7B2%7D&id=XBxsS)

样本矩：样本阶原点矩：

样本阶中心矩：%7D%5E%7Bk%7D%2Ck%20%3D%201%2C2%2C%5Ccdots#card=math&code=B%7Bk%7D%20%3D%20%5Cfrac%7B1%7D%7Bn%7D%5Csum%7Bi%20%3D%201%7D%5E%7Bn%7D%7B%28X_%7Bi%7D%20-%20%5Coverline%7BX%7D%29%7D%5E%7Bk%7D%2Ck%20%3D%201%2C2%2C%5Ccdots&id=Dq3jK)

2.分布

分布：#card=math&code=%5Cchi%5E%7B2%7D%20%3D%20X%7B1%7D%5E%7B2%7D%20%2B%20X%7B2%7D%5E%7B2%7D%20%2B%20%5Ccdots%20%2B%20X%7Bn%7D%5E%7B2%7D%5Csim%5Cchi%5E%7B2%7D%28n%29&id=UHz9K)，其中相互独立，且同服从#card=math&code=N%280%2C1%29&id=UdM8X)

分布：#card=math&code=T%20%3D%20%5Cfrac%7BX%7D%7B%5Csqrt%7BY%2Fn%7D%7D%5Csim%20t%28n%29&id=jCA1m) ，其中%2CY%5Csim%5Cchi%5E%7B2%7D(n)%2C#card=math&code=X%5Csim%20N%5Cleft%28%200%2C1%20%5Cright%29%2CY%5Csim%5Cchi%5E%7B2%7D%28n%29%2C&id=sN8pf)且， 相互独立。

分布：#card=math&code=F%20%3D%20%5Cfrac%7BX%2Fn%7B1%7D%7D%7BY%2Fn%7B2%7D%7D%5Csim%20F%28n%7B1%7D%2Cn%7B2%7D%29&id=FvEsi)，其中%2CY%5Csim%5Cchi%5E%7B2%7D(n%7B2%7D)%2C#card=math&code=X%5Csim%5Cchi%5E%7B2%7D%5Cleft%28%20n%7B1%7D%20%5Cright%29%2CY%5Csim%5Cchi%5E%7B2%7D%28n_%7B2%7D%29%2C&id=ffqg4)且，相互独立。

分位数：若%20%3D%20%5Calpha%2C#card=math&code=P%28X%20%5Cleq%20x%7B%5Calpha%7D%29%20%3D%20%5Calpha%2C&id=O1Whw)则称为的分位数

3.正态总体的常用样本分布

(1) 设为来自正态总体#card=math&code=N%28%5Cmu%2C%5Csigma%5E%7B2%7D%29&id=WxIZg)的样本，

%7D%5E%7B2%7D%2C%7D#card=math&code=%5Coverline%7BX%7D%20%3D%20%5Cfrac%7B1%7D%7Bn%7D%5Csum%7Bi%20%3D%201%7D%5E%7Bn%7DX%7Bi%7D%2CS%5E%7B2%7D%20%3D%20%5Cfrac%7B1%7D%7Bn%20-%201%7D%5Csum%7Bi%20%3D%201%7D%5E%7Bn%7D%7B%7B%28X%7Bi%7D%20-%20%5Coverline%7BX%7D%29%7D%5E%7B2%7D%2C%7D&id=rvFYa)则：

1. %7B%5C%20%5C%20%7D#card=math&code=%5Coverline%7BX%7D%5Csim%20N%5Cleft%28%20%5Cmu%2C%5Cfrac%7B%5Csigma%5E%7B2%7D%7D%7Bn%7D%20%5Cright%29%7B%5C%20%5C%20%7D&id=a7EHU)或者#card=math&code=%5Cfrac%7B%5Coverline%7BX%7D%20-%20%5Cmu%7D%7B%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D%5Csim%20N%280%2C1%29&id=huut3)
2. S%5E%7B2%7D%7D%7B%5Csigma%5E%7B2%7D%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Csigma%5E%7B2%7D%7D%5Csum%7Bi%20%3D%201%7D%5E%7Bn%7D%7B%7B(X%7Bi%7D%20-%20%5Coverline%7BX%7D)%7D%5E%7B2%7D%5Csim%5Cchi%5E%7B2%7D(n%20-%201)%7D#card=math&code=%5Cfrac%7B%28n%20-%201%29S%5E%7B2%7D%7D%7B%5Csigma%5E%7B2%7D%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Csigma%5E%7B2%7D%7D%5Csum%7Bi%20%3D%201%7D%5E%7Bn%7D%7B%7B%28X%7Bi%7D%20-%20%5Coverline%7BX%7D%29%7D%5E%7B2%7D%5Csim%5Cchi%5E%7B2%7D%28n%20-%201%29%7D&id=RqTL0)
3. %7D%5E%7B2%7D%5Csim%5Cchi%5E%7B2%7D(n)%7D#card=math&code=%5Cfrac%7B1%7D%7B%5Csigma%5E%7B2%7D%7D%5Csum%7Bi%20%3D%201%7D%5E%7Bn%7D%7B%7B%28X%7Bi%7D%20-%20%5Cmu%29%7D%5E%7B2%7D%5Csim%5Cchi%5E%7B2%7D%28n%29%7D&id=E3oMI)

4)#card=math&code=%7B%5C%20%5C%20%7D%5Cfrac%7B%5Coverline%7BX%7D%20-%20%5Cmu%7D%7BS%2F%5Csqrt%7Bn%7D%7D%5Csim%20t%28n%20-%201%29&id=JRCyQ)

4.重要公式与结论

(1) 对于#card=math&code=%5Cchi%5E%7B2%7D%5Csim%5Cchi%5E%7B2%7D%28n%29&id=HEtYV)，有)%20%3D%20n%2CD(%5Cchi%5E%7B2%7D(n))%20%3D%202n%3B#card=math&code=E%28%5Cchi%5E%7B2%7D%28n%29%29%20%3D%20n%2CD%28%5Cchi%5E%7B2%7D%28n%29%29%20%3D%202n%3B&id=uXRQl)

(2) 对于#card=math&code=T%5Csim%20t%28n%29&id=ryyjs)，有%20%3D%200%2CD(T)%20%3D%20%5Cfrac%7Bn%7D%7Bn%20-%202%7D(n%20%3E%202)#card=math&code=E%28T%29%20%3D%200%2CD%28T%29%20%3D%20%5Cfrac%7Bn%7D%7Bn%20-%202%7D%28n%20%3E%202%29&id=fk2KY)；

(3) 对于#card=math&code=F%5Ctilde%7B%5C%20%7DF%28m%2Cn%29&id=CGvpC)，有 %2CF%7Ba%2F2%7D(m%2Cn)%20%3D%20%5Cfrac%7B1%7D%7BF%7B1%20-%20a%2F2%7D(n%2Cm)%7D%3B#card=math&code=%5Cfrac%7B1%7D%7BF%7D%5Csim%20F%28n%2Cm%29%2CF%7Ba%2F2%7D%28m%2Cn%29%20%3D%20%5Cfrac%7B1%7D%7BF%7B1%20-%20a%2F2%7D%28n%2Cm%29%7D%3B&id=u3U8O)

(4) 对于任意总体，有 %20%3D%20E(X)%2CE(S%5E%7B2%7D)%20%3D%20D(X)%2CD(%5Coverline%7BX%7D)%20%3D%20%5Cfrac%7BD(X)%7D%7Bn%7D#card=math&code=E%28%5Coverline%7BX%7D%29%20%3D%20E%28X%29%2CE%28S%5E%7B2%7D%29%20%3D%20D%28X%29%2CD%28%5Coverline%7BX%7D%29%20%3D%20%5Cfrac%7BD%28X%29%7D%7Bn%7D&id=lrf0h)