大學(xué)概率論與數(shù)理統(tǒng)計(jì)公式全集

大學(xué)概率論與數(shù)理統(tǒng)計(jì)公式全集UniversityProbabilityandMathematicalStatisticsCompleteFormulaCollectionI.RandomEventsandProbability1.RandomeventsandtheirprobabilitiesLawofoperationnames:-Commutativelaw-Associativelaw-Distributivelaw-DeMorgan'slaw2.DefinitionandcalculationofprobabilityFormulanames:-Inverseprobabilityformula-Additionformula-Conditionalprobabilityformula-Multiplicationformula-Totalprobabilityformula-Bayes'formula(inverseprobabilityformula)-Bernoulliformula-FormulafortwoindependenteventsP(AB)=P(A)P(B)Expression:A+B=B+A(A+B)+C=A+(B+C)=A+B+CA(B±C)=AB±ACA+B=ABAB=BA(AB)C=A(BC)=ABCA+(BC)=(A+B)(A+C)AB=A+BFormulaexpression:P(A)=1-P(A)P(A+B)=P(A)+P(B)-P(AB)P(BA)=P(AB)/P(A)P(AB)=P(A)P(BA)nP(AB)=P(B)P(AB)iiP(B)=ΣP(A)P(BA)i=1P(AjB)=P(Aj)P(BAj)/ΣP(A)P(BA)jii=1∞P(n(k))=Cn(p(1-p)n-k)k=1,2,...,nP(BA)=P(B)P(BA)=P(BA)P(BA)+P(BA)=1P(BA)+P(BA)=1II.RandomVariablesandTheirDistributions1.PropertiesofdistributionfunctionsP(X≤b)=F(b)P(a<X≤b)=F(b)-F(a)2.DiscreterandomvariablesDistributionnames:-Uniformdistribution-Bernoullidistribution-Binomialdistribution-Poissondistribution-Geometricdistribution-Hypergeometricdistribution3.ContinuousrandomvariablesDistributionnames:-Uniformdistribution-Exponentialdistribution-Normaldistribution-StandardnormaldistributionDensityfunction:-Uniformdistribution:f(x)=1/(b-a),a<x<b-Exponentialdistribution:f(x)=λe-λx,x≥0-Normaldistribution:other-Standardnormaldistribution:f(x)=(1/√(2π))e-(x^2)/2分布，記為T~t(n)。性質(zhì)：①E[T(n)]=0，當(dāng)n>1時(shí)，D[T(n)]=n/(n-2)。②若X~N(,1),Y~2(n)，且X與Y獨(dú)立，則T=X/(Y/n)^(1/2)~t(n)。(3)F分布：設(shè)隨機(jī)變量X~2(m),Y~2(n)，且X與Y獨(dú)立，則隨機(jī)變量：F=X/m/(Y/n)~F(m,n)。性質(zhì)：①E[F(m,n)]=n/(n-2)，當(dāng)n>2時(shí)，D[F(m,n)]=2n^2(m+n-2)/(m(n-2)^2(n-4))。②若X~N(0,1),Y~2(n)，且X與Y獨(dú)立，則(T(n))^2=X^2/Y~F(1,n)。分布：記為$T\simt(n)$，其性質(zhì)為：①$E[t(n)]=0$，$D[t(n)]=\frac{1}{n-2}$，其中$n>2$；②當(dāng)$n\to\infty$時(shí)，$t(n)$的極限分布為$N(0,1)$。F分布：設(shè)隨機(jī)變量$U\sim\chi^2(n_1)$，$V\sim\chi^2(n_2)$且$U$與$V$獨(dú)立，則隨機(jī)變量$F(n_1,n_2)=\frac{U/n_1}{V/n_2}$服從自由度為$(n_1,n_2)$的F分布，記為$F\simF(n_1,n_2)$。其中，若$X\simF(m,n)$，則$X^{-1}\simF(n,m)$。參數(shù)估計(jì)：參數(shù)估計(jì)是指用樣本統(tǒng)計(jì)量估計(jì)總體參數(shù)的過程。估計(jì)總體參數(shù)$\theta$的估計(jì)量為$\theta(X_1,X_2,\ldots,X_n)$，相應(yīng)的$\theta(X_1,X_2,\ldots,X_n)$為總體$\theta$的估計(jì)值。當(dāng)總體是正態(tài)分布時(shí)，未知參數(shù)的矩估計(jì)值等于未知參數(shù)的最大似然估計(jì)值。點(diǎn)估計(jì)中的矩估計(jì)法：對(duì)于離散型樣本，其均值為$X=\frac{1}{n}\sum\limits_{i=1}^nX_i$；對(duì)于離散型參數(shù)，其期望為$E(X)=\frac{1}{n}\sum\limits_{i=1}^nX_i^2$；對(duì)于連續(xù)型樣本，其均值為$X=\int_{-\infty}^{+\infty}xf(x,\theta)dx$。點(diǎn)估計(jì)中的最大似然估計(jì)法：設(shè)$X_1,X_2,\ldots,X_n$取自$X$的樣本，其概率密度為$f(x,\theta)$（或概率為$P(X=X_i)=P(\theta)$），則可得到概率密度函數(shù)$f(x_1,x_2,\ldots,x_n,\theta)=\prod\limits_{i=