作业九

151220129 计科 吴政亿

习题六 第3题

1. S2=1n−1∑ni=1(Xi−X¯¯¯)2
   =1n−1∑ni=1[X2i+X¯¯¯2−2XiX¯¯¯]
   =1n−1(∑ni=1X2i+nX¯¯¯2)−2X¯¯¯1n−1∑ni=1Xi
   =1n−1(∑ni=1X2i+nX¯¯¯2)−2nn−1X¯¯¯2=1n−1[∑ni=1X2i−nX¯¯¯2]
2. ∵E((n−1)S2σ2)=n−1=(n−1)σ2E(S2)
   ∴E(S2)=σ2

习题六 第8题

E(Y)=∑ni=1D(Xi+Xn+i−2X¯¯¯)+∑ni=1E(Xi+Xn+i−2X¯¯¯)2
=∑2ni=1D(Xi)−4∑ni=1cov(Xi,X¯¯¯)−4∑ni=1cov(Xn+i,X¯¯¯)+4∑ni=1D(X¯¯¯)+0
=2nσ2−4n[cov(Xi,Xi2n)+cov(Xn+i,Xn+i2n)]+2σ2=2(n−1)σ2

习题六 第9题

∵(n−1)S2/σ2~χ2(n−1),X¯¯¯~N(μ,σ2n),Xn+1~N(μ,σ2)
∴Xn+1−X¯¯¯~N(0,σ2n+1n)
∴Xn+1−X¯Snn+1−−−√=nn+1−−−√Xn+1−X¯(n−1)S2/(n−1)√~N(0,n+1n)χ2(n−1)/(n−1)√nn+1−−−√
~N(0,1)χ2(n−1)/(n−1)√~t(n−1)

习题六 第10题

1. X¯¯¯~N(12,45),P(X¯¯¯>13)=P(X¯−120.8√>1.25−−−−√)=1−Φ(1.1180)=0.131

2. P(min1≤i≤5Xi<10)=1−P(Xi≥10)5=1−Φ(1)5=0.5785

3. P(max1≤i≤5Xi>15)=1−P(Xi≤15)5=1−Φ(1.5)5=0.2923

习题六 第11题

设联合样本均值为Z,方差为S2则有Z¯¯¯=n1X¯+n2Y¯n1+n2
∵S21=1n1−1∑n1i=1X2i−n1n1−1X¯¯¯2,    ∴∑n1i=1X2i=(n1−1)S21+n1X¯¯¯2
S2=1n1+n2−1(∑n1i=1X2i+∑n2i=1Y2i)−n1+n2n1+n2−1Z¯¯¯2
=(n1−1)S21+n1X¯2+(n2−1)S22+n2Y¯2n1+n2−1−n1+n2n1+n2−1(n1X¯+n2Y¯n1+n2)2

第二题

E((X1+X2)(X1−X2))=E(X21−X22)=EX21−EX22
=(EX1−EX2)(EX1+EX2)=E(X1+X2)E(X1−X2)
故(X1+X2)2,(X1−X2)2独立。
∵X1+X2~N(0,2σ2),∴(X1+X2)22σ2~χ(1),同理(X1−X2)22σ2~χ(1)
故(X1+X2)2(X1−X2)2~χ(1)χ(1)~F(1,1)