1、 5 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) =s、+ s Z T wV,C3BwV I. 预备公式 1 iCin=nCi-1n-1(n1), F 9 T V . 预备公式 2 D=E2 -(E)2, n. 预备公式 3 C0nCkm+C1nCk-1m +C2nCk-2m +CknC0m=Ckn+m(n,m, k N,kn,km), T(1+x)n+m=(1+x)n(1+ x)m=Z 7 Txk M V. B、=s L, T? 3 q (p(? 3 qq,p+q=1), * nQ LT? 3 Q
2、 qs: 0 1 2 3 n-1 n P C0nqn C1npqn-1 C2np2qn-2 C3np3qn-3 Cn-1n pn-1qCnnpn E()= n i=0 iCinpiqn-i = n i=1 iCinpiqn-i = n i=1 nCi-1n-1piqn-i( ! T1 V) =np n i=1 Ci-1n-1pi-1qn-i =np(p+q)n-1 =np. V()=E2 -(E)2 = n i=0 i2Cinpiqn-i-n2p2 = n i=1 niCi-1n-1piqn-i-n2p2 =n n i=1 (i-1)Ci-1n-1piqn-1+n n i=1 Ci-1n-1p
3、iqn-i-n2p2 =p2n(n-1) n i=2 Ci-2n-2pi-2qn-i+ np n i=1 Ci-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). =、+ s B Nq, Mq , |n q , X qs: X 0 1 2 3 l-1 l P C0MCnN-M CnN C1MCn-1N-M CnN C2MCn-2N-M CnN C3MCn-3N-M CnN Cl-1M Cn+1-lN-M CnN ClMCn-lN-M CnN l=min(n,M). E
4、(x)= l i=0 iC i MC n-i N-M CnN =MCn N l i=1 Ci-1M-1Cn-iN-M =MCn N Cn-1N-1( ! T3 V) =nMN. V(x)= l i=0 i2 C i MC n-i N-M CnN - Mn N 2 =MCn N l i=1 iCi-1M-1Cn-iN-M-MnN 2 =MCn N l i=1 (i-1)Ci-1M-1Cn-iN-M+ MCn N l i=1 Ci-1M-1Cn-iN-M-MnN 2 =MCn N (M-1) l i=2 Ci-2M-2Cn-iN-M+ MCn N l i=1 Ci-1M-1Cn-iN-M-MnN 2 =MCn N (M-1)Cn-2N-2+MCn N Cn-1N-1 -MnN 2 =nMN 1-MN 1-n-1N-1 .
