概率论与数理统计知识点总结

《概率论与数理统计》第一章随机事件及其概率§§=nn1：nn...nnn1n(n(n2)...1!P()!nn343i？44443641432AP(A)1243648C2A2343P(A2)36964163C3344P(A3)416416P(P()§B∪））、、n1)+))A、、n：P（A）BA与－A1A2...AnA1A2...AnA1A2...AnA1A2...An§条件概率公式：P(AB)P(AB)）P(B)P()P=B/）PP全概率与逆概率公式：nP(B)P(Aii1)P(B/Ai)P(AB)P(A/B)P(B)(iin)i§P(AB)P()P(B)事件的独立性：贝努里公式（nBAABA与A与、与、与P(A1A2...An)1P(A1A2...An)第二章随机变量及其分布2p注意：应符合性质——、pk0、pkk12例中有在其时取只，表示3k3C5345：pkF(x)Px1/10pkxkx3/106/10=x(x)dx，则(x)(x)0(x)dx1a}a}F(b)F(a)(x)dxba第三章随机变量数字特征3？Exkkpkx1x2…xkpkη=)py11py22……pykk510122pk11133510求：⑴1，2E。510122pk111335103η2101225η101443η21012pk1113351025η10144pk11133510－1η（－）111333

=××××+×

521113253

η××××+×

544D？？DEE22=－E(E)22(xp)2==kkkE2=x2kkpk202pkE和DED－E=E常用分布的均值与方差布kCpqkknkn(kn)npnpqp布(x)x()..，为常数1(x)2,e2μ2布λλ5布112E和D的性质（EDE(c)cD(c)0若、独立，则E)EED)DDE)E()cD(c)c2D§ˆlimˆ}1ˆP{，则称nˆ)Eˆˆˆ1和2ˆˆ)()12ˆˆ12§ˆ(x，...11n)及ˆ(x，...21n)，对于给定的6ˆ(x，ˆ(x，)......1121nn)}1ˆˆ1，2的－ˆˆ1和2的－－E2U0)－2U；②x=1nnxi1ixxd=UnX~N(,0.09)4α2α141②XX13.412.813.2)1344ii1U=4n（XX2nt(n；x=1nnxi1i和1sn(xx2n1ii1)2t(nsnxx2n7(n2(n2和2

21n1xns(Xx)22X=和nin1ii1i1(ns2(ns22(n2(n22之(n22=0.022(9)22(n=20.98(9)②X=1i1xi1=(482493...469)10s(11Xx)2=[(457.5482)2+(457.5493)2(457.5469)2]299ii1(ns29s292(n=2=2(ns29s292(n=2=2第六章假设检验8值检验类型⑵：未知方差2μ:=0(H0TXs/n~t(n；nt(nT0Xs/n出Xs；若t(n，则接受H00Tt(n，则拒绝H005寸2）寸2:=0TXs/n~t(n∵t0.05(4)X=15...=X122T...(543]0=4s/n/5μ，检验总体方差2H:=00(0(n2(ns22；n22(n(n2和；29Xs2(n0(ns22；(n22(n<<2022(n2，。）H:(ns22(n2∵(n2(9)22=0.97522(n=20.025(9)X=11...570)=[(575.292...(575.2570)2]∴2(n0975.736410.6520.975(9)2(n<0(9)20.025。第1章Pnm!(mn从mnCnm!!(mn从mn理10件…表示事件，它们是1ABABABABBAABAA=B。A。ABA与BA-BA-ABABABAA与BABAAAAAii1i1AiABAB，ABAB设AAA，A2PiAP(Ai)i1i111A,12P1n)P2，)Pn)1n。A，它是由,P(A)=12=m))P)P)P1212mm)mnP()L()L()L当当当BA当B设P(AB)P()AP(AB)

件BP(B/)。P()BP(AB)P()P(B/)AA12n12P(A)P(A2|)P(A3|A2)AAAAn)P(AP(An|A2…2…111An)。n)。P(AB)P(A)P(B)ABABP(A)0A、BP(B|)P(AB)P()P(B)P(B)P()P()性A、BA与B、A与B、A与BØØ12设nB,B2,,BnB,B2,,BnP(B)0(i,n)i，nABii1，P(A)P(B)P(A|B)P(B2)P(A|B2)P(B)P(A|B)nn。B1，B2n及AB1，B2BnP(Bi)in，则nABi1，P(A)0，iP(B)P(A/B)P(B/)iiinP(B)P(A/B)jjj1P(B)i12nP(Bii/A)i12，nnAAnAAAn用pAA1pqP(k)n表示nAk(0kn)P(k)Cnknpkqnk，k,n。13第二章随机变量及其分布律XXkkkkXXx,x2,,x,|kP(Xx)p,p2,,p,kkpk0，kpk，k11。设F(x)Xf(x)xF(x)f(x)dxx，Xf(x)X4f(x)0f(x)dx1。P(xXx)F(x)F(x)f(x)dx21221x10P(Xx)P(xXxdx)f(x)f(x)P(Xx)pkk14设XxF(x)P(Xx)XP(aXb)Fb)F(a)X(a,b]数F(x)0F(x)x；F(x)x1x2F()F(x2)；F()limF(x)0，F()limF(x)1；xxF(xF(x)F(x)P(Xx)F(x)F(x0)。F(x)pkxx；F(x)xf(x)dx。在nApAXX,n。P(Xk)P(k)Cknnpkqnk，q1p,0pk,n，XnpX~B(n,p)。当n1P(Xk)pkqk，k0.115XP(Xk)kk!e，0，k，XX~()P(Xk)qk1p,kXpXf(x)1ba1,f(x)baX~，，xa,baF(x)f(x)dxx，。,x当2x112xxP(xXx)21。ba1216ex,x0,f(x)x0,0XXF(x)1e,x0,x0,。xnexdx!0X1(x)22，x，f(x)

e0X的X~N(,)。2f(x)f(x)x1当xf()若X~N(,)2X1(t)F(x)e02x2dt12X~N((x)x21e2，x，1et2x(x)2dt。(x)1。2XX~N(,)~N(。2P(x1Xx2xx)21。17数P(X＝；P(X＝。X数x,x2,,x,XnP(Xx),p2,,p,in，Yg(X)yg(x)iig(xg(x2,g(xYn，PYy)p,p2,,p,ing(x)piig(x)iXfYFXYfY（当:是18设(xi,y)(i,j)，j=(xi,yj)pij,{(X,Y)(xi,yj)}p(i,j)为X和YYXy1y2…yj…x1p11p12…p1j…x2p21p22…p2j…xipi1……pijpijpijpij1.ij对于二维随机向量(X,Y)f(x,y)(x,y)有{(X,Y)}f(x,y)dxdy,Df(x,y)为X和Yf(x,y)1.19（(Xx,Yy)(XxYy)（F(x,y){Xx,Y}和Y,12)|X1)x,Y2)}0F(x,y)x和y当xy)21212121x和yF(x,y)F(xyF(x,y)F(x,y0);F(,)F(,y)F(x,)F(,)1.（x1x，y21y，2≤≤F(x≤≤F(xy22)F(xy)F(xy2112)F(xy)011P(X，Yy)P(xXx，yYydy)f(，y)dxdyXPi•P(Xxi)pijji,j)；YP•jPYyj)pi(i,j)。Xf(x)Xf(x,y)；YfY(y)f(x,y).20X=xiPYyj|Xxi)ppiji•；Y=yjP(Xxi|Yy)jppij•j,f(x|y)f(x,y)fY(y)；f(y|x)f(x,y)f(x)XXY性ppi•p•jXY布f(x,y)11212x22(x)(y)y2111222(12),e1122若X,X,…X12mn12mnX与YX与Y和211SDf(x,y)(x,y)D其他SDDDy1D1O1x图y1D2O2x1图ydD3cOabx图22f(x,y)11212ex22(xy)y21112222(1)1122,12,10,0,|1是52,,2212,12,即,),Y~N(22112,2,),Y~N(22112,2)FZ(z)P(Zz)P(XYz)f(x,zxdx12,2212nC，2C22iiiiii…),XX若X12nFx1(x，Fx2(x)Fxn(x)…F(x)Fx1(x)•Fx2(x)Fxn(x)F(x)1Fx1(x)]•Fx2(x)]Fxn(x)]123设X设X维随机Xx，kkE(X)xf(x)dx变量的nE(X)xkk1pk数字特征nEY)g(xkk1)pkEY)g(x)f(x)dx，D(X)[xkkE(X)]p2kD(X)[xE(X)]