二项分布超几何分布数学期望与方差公式的推导

　　　　　　ZHUANTIYANJIU108　　　2010.11二项分布、超几何分布数学期望与方差公式的推导●韩晓东　(江苏省淮阴中学　223002)　　、,.预备公式1iCin=nCi-1n-1(n≥1),.预备公式2Dξ=Eξ2-(Eξ)2,.预备公式3C0nCkm+C1nCk-1m+C2nCk-2m+…+CknC0m=Ckn+m(n,m,k∈N＊,k≤n,k≤m),(1+x)n+m=(1+x)n(1+x)mxk.、,p(q,p+q=1),nξ:ξ0123…n-1nPC0nqnC1npqn-1C2np2qn-2C3np3qn-3…Cn-1npn-1qCnnpn　　E(ξ)=∑ni=0iCinpiqn-i=∑ni=1iCinpiqn-i=∑ni=1nCi-1n-1piqn-i(1)=np∑ni=1Ci-1n-1pi-1qn-i=np(p+q)n-1=np.V(ξ)=Eξ2-(Eξ)2=∑ni=0i2Cinpiqn-i-n2p2=∑ni=1niCi-1n-1piqn-i-n2p2=n∑ni=1(i-1)Ci-1n-1piqn-1+n∑ni=1Ci-1n-1piqn-i-n2p2=p2n(n-1)∑ni=2Ci-2n-2pi-2qn-i+　np∑ni=1Ci-1n-1pi-1qn-i-n2p2=p2n(n-1)(p+q)n-2+np(p+q)n-1-n2p2=p2n(n-1)+np-n2p2=np-p2n=np(1-p).、N,M,n,X:X0123…l-1lPC0MCnN-MCnNC1MCn-1N-MCnNC2MCn-2N-MCnNC3MCn-3N-MCnN…Cl-1MCn+1-lN-MCnNClMCn-lN-MCnN　　l=min(n,M).E(x)=∑li=0iCiMCn-iN-MCnN=MCnN∑li=1Ci-1M-1Cn-iN-M=MCnNCn-1N-1(3)=nMN.V(x)=∑li=0i2CiMCn-iN-MCnN-MnN2=MCnN∑li=1iCi-1M-1Cn-iN-M-MnN2=MCnN∑li=1(i-1)Ci-1M-1Cn-iN-M+　MCnN∑li=1Ci-1M-1Cn-iN-M-MnN2=MCnN(M-1)∑li=2Ci-2M-2Cn-iN-M+　MCnN∑li=1Ci-1M-1Cn-iN-M-MnN2=MCnN(M-1)Cn-2N-2+MCnNCn-1N-1-MnN2=nMN1-MN1-n-1N-1.