所以成绩在60到84的概率为
P(60?X?84)?P(60-72X-?84-72??) 12?12 ??(1)-?(-1) ?2?(1)-1 ?2?0.8413-1 ?0.6826
4.12E(X2)?0?0.4?12?0.3?22?0.2?32?0.1?2
E(5X2?4)?4?0.4?(5?12?4)?0.3?(5?22?4)?0.2?(5?32?4)?0.1?14
4.13解：
E(Y)?E(2X)??2xe?xdx?2?xd(?e?x)?2[?xe?x|??e?xdx]0000?????2(?e)|?2?x0?
???11E(Y)?E(e?2X)??e?2xe?xdx??e?3xdx??e?3x|? 000334.14
4?R3解：V?3
?1a?x?b? f(x)??b?a 其它??0设球的直径为X,则：
4?(E(V)?E(X3)2)?E(?X3)=b?x31dx???1?1x4b??(b?a)(b2?a2)?a6b?a6b?a4|a24364.15参看课本后面231页答案 4.16 解:
fx(x)????f(x,y)dy??012ydy?4x3
??x2fy(y)??f(x,y)dy??12ydx?12y?12y??y????1223
E(X)????fx(x)?xdx??104xdx?344 54E(Y)??????fy(x)?ydy??12y?12ydy?013 51xE(XY)?0?y?x?1??f(x,y)xydxdy?0?y?x?1??12xydxdy??5300?12xydydx?31 2E(X)??f(x)?xdx????2??2104xdx?42 35E(Y)??f(y)?ydy??12y?12ydy?2??0??212 5E(X2?Y2)?E(X)?E(Y)?2216 15 4.17解
∵X与Y相互独立， ∴
1?21?E(XY)?E(X)E(Y)??x2xdx?ye5?ydy?(x3|)?yd(?e5?y)05305???222??(?ye5?y|??e5?ydy)??[5?(?e5?y)|]??(5?1)?4555 3334.18，4.19，4.20参看课本后面231，232页答案
4.21设X表示10颗骰子出现的点数之和，Xi(i?1,2,L10)表示第
i颗骰子出现的点数，则X??Xi，且X1,X2,LX10是
i?110独立同分布的，又E(Xi)?1?1?2?1?L66121?6??66
所以E(X)?E(?Xi)??E(Xi)?10?i?1i?1101021?35 64.22参看课本后面232页答案
4.23E(X2)?0?0.4?12?0.3?22?0.2?32?0.1?2
D(X)?E(X2)?[E(X)]2?2?12?1 E(Y2)?0?0.3?12?0.5?22?0.2?32?0?1.3 D(Y)?E(Y2)?[E(Y)]2?1.3?0.92?0.49
4.24
E(X2)??x202424111111114xdx??x2(?x?1)dx?x4|?[?x4?x3]|?1?? 224416016333D(X)?E(X2)?[E(X)]2?142?4? 334.25
?1?x?1?1?11?xy?1?x?1dy??=?2 fX(x)????14
其它???0?0其它1112xdx?[?xdx]2 ?12?121Var(X)?E(X2)?[E(X)]2??1111111??x3|??x2|? 23?122?13
?1?y?1?1?11?xy?1?y?1dx??=fY(y)????14 ?2
其它???0?0其它1112Var(Y)?E(Y)?[E(Y)]??ydy?[?ydy]2 ?12?122211111111??y3|??y2|? 23?122?134.26因为X~N(0,4),Y~U(0,4)所以有Var(X)=4 Var(Y)= 故：Var(X+Y)=Var(X)+Var(Y)=4+=
434316 34?28 3Var(2X-3Y)=4Var(X)+9Var(Y)= 4?4?9?4.27参看课本后面232页答案 4.28E(Z)?E(?X1?X2?L?XnXXX)?E(1)?E(2)?L?E(n)
nnnn1111E(X1)?E(X2)?L?E(Xn)???n?? nnnnD(Z)?D(X1?X2?L?XnXXX)?D(1)?D(2)?L?D(n)
nnnn11112?2?2E(X1)?2E(X2)?L?2E(Xn)?2??n?nnnnn
后面4题不作详解
第五章 极限理
5.3
解：用Xi表示每包大米的重量，，则E(Xi)???10,D(Xi)??2?0.1
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