大學(xué)概率論與數(shù)理統(tǒng)計(jì)公式總結(jié),期末考試不掛科的法寶

概率公式整理AAAA(AB)A

AAAA(AB)

ABABA(AB)反演律：ABAB ABAB nAnA nAnAi i i ii1 i1 i1 i1概率的定義及其計(jì)算：P(A)1P(A) 若AB P(BP(B)P(對任意兩個事件A,B,有P(BA)P(B)P(AB)加法公式：對任意兩個事件A,B,有P(AB)P(P(B)P(AB) P(AB)P(P(B)Pn

A)

P(A)

P(AA)

nP(AAA)()n1P(AA

A)ii1 i1

i i 1ijn

1ijkn

i j k

1 2 n條件概率P(AB) 乘法公式P(AB)P(A(P( P( P(AA1

A)P(A)PA An 1 2 1

A AAn 1

An1(P(AA1 2n

An1n

)0)

P(AB)

P(B)P(AB)PA)

P(AB)

P(B)PA BBayesP(B

A)

k k ki1

i i i1

k P(A)

P(B)P(AB)

P(aXb)P(Xb)P(Xa)F(b)F(a)

i ii15．離散型隨機(jī)變量(1) 0–1分布P(Xk)pk(1p)1k, k二項(xiàng)分布B(n,p)若P(A)=p P(Xk)Ckpkp)nk, knnlimCkpk(1p

)nk

ekPossion定理limnpn n

0 有

n n n

n k!k0,1,2,Poisson分布P() P(Xk)ekk!

, k

1 , axb

xa連續(xù)型隨機(jī)變量(1) 均勻分布U(a,b)f(x)ba F(x) ,

ba 1(2)指數(shù)分布E() f(x)ex, x0 F(x) ,

x0 , 其他

, x0(3)正態(tài)分布N(,2) f(x)

(x)221e 22 x F(x)21

1 x22

(t)2e 22dtN(0,1)標(biāo)準(zhǔn)正態(tài)分布

(x) e

2x2 xx

(x)

1 x 12122

t22dt x多維隨機(jī)變量及其分布 二維隨機(jī)變量(X,Y)的分布函數(shù) F(x,y)xyf(u,v)dvdu邊緣分布函數(shù)與邊緣密度函數(shù)F

(x)

fu,v)dvdu f

(x)f(x,v)dvX X F(y)yf(u,v)dudv f(y)fu,y)duY Y 1, (x,y)G連續(xù)型二維隨機(jī)變量 (1) 區(qū)域G上的均勻分布，U(G)f(x,y)A(2)二維正態(tài)分布1

1 (x

)22(x1

)(y2

, 其他1 2 2 )(y2)1 2 2 f(x,y)

2 1 2

e12

2(12)12

2 x, yf(x,y)fX

fY

(yx)

f (x)0 fX

)fXY

(xy)

f(y)0Yf (x)f(x,y)dyf

(xy)f

(y)dy

(y)f(x,y)dxf

(yx)

(x)dxX

f(x,y)

XYf (yx)

Y(x)

Y

f(x,y)

YXf (xy)

X(y)f (xy)

YX X

f (yx) XY YXY fY

(y)

f(y)Y

YX fX

(x)

f (x)X隨機(jī)變量的數(shù)字特征 數(shù)學(xué)期望E(X)xpk kk1

E(X)xf(x)dx隨機(jī)變量函數(shù)的數(shù)學(xué)期望X的k階原點(diǎn)矩E(Xk) X的k階絕對原點(diǎn)矩E(|X|k)X的k階中心矩E((XE(X))k) X的方差E((XE(X))2)D(X)X,Y的k+l階混合原點(diǎn)矩E(XkYl) X,Y的k+l階混合中心矩

(XE(X))kE(Y))lX,Y的二階混合原點(diǎn)矩E(XY) X,Y的二階混合中心矩 X,Y的協(xié)方差XE(XE(Y))(XE(XE(Y))X,Y的相關(guān)系數(shù)E

D(X) D(Y)

XY X的方差D(X)=E((X-E(X))2) E(X2)E2(X)協(xié)方差cov(X,YXEXE(Y