1、練習5.11、仿照本節的例子，分別畫出二項分布B(20,0.7)的分布規律和分布函數的圖形，通過觀察圖形，進一步理解二項分布的性質。解：分布規律編程作圖：>> x=0:1:20;y=binopdf(x,20,0.7);>> plot(x,y,'*') 圖像：0.20.180.160.14 0.12 y 0.10.080.06+ * 七*七+-0.04 0.02005101520x分布函數編程作圖：>> x=0:0.01:20;>>y=binocdf(x,20,0.7)>> plot(x,y) 圖像：10.90.80.7

2、0.6y 0.50.4 0.3 - 0.20.1 -1015x200 L05觀察圖像可知二項分布規律圖像像一條拋物線，其分布函數圖像呈階梯狀。2、仿照本節的例子，分別畫出正態分布N(2,52 )的概率密度函數和分布函數的圖形，通過觀察圖形，進一步理解正態分布的性質。解：概率密度函數編程作圖： >> x=-10:0.01:10;>> y=normpdf(x,2,5);>> plot(x,y)圖像：0.080.070.060.05y 0.040.030.020.010 L-10-510分布函數編程作圖：>> x=-10:0.01:10;>>

3、;>>y=normcdf(x,2,5); plot(x,y)圖像:0.90.80.70.60.40.30.20.1-510y 0.50 -10x觀察圖像可知正態分布概率密度函數圖像像拋物線，起分布函數圖像呈遞增趨勢。3、設X N(0,1 ),通過分布函數的調用計算P1<X <仆，P2<X <2,P13 :二 X ：二3).解：編程求解：>>x1=normcdf(1)-normcdf(-1),x2=normcdf(2)-normcdf(-2),x3=normcdf(3)-normcdf(-3)x1 = 0.6827x2 = 0.9545x3 = 0

4、.9973即：P1 <X <1=0.6827 , P2 <X <2= 0.9545 , P-3<X <3= 0.9973.4、設X B(20,0.7),通過分布函數的調用計算PX =10與PX <10.解：編程求解：>> x1=binopdf(10,20,0.7),x2=binocdf(10,20,0.7)-binopdf(10,20,0.7)x1 = 0.0308x2 = 0.0171即：P僅=10=0.0308 , PtX <10) = 0.01715、設 XP(8)求：(1) p!x <4;(2) P2<XE5.解：

5、(1)編程求解：>> p=poisscdf(4,8) p = 0.0996即：P ：X ,4? - 0.0996(2)編程求解：>> p=poisscdf(5,8)-poisscdf(2,8) p = 0.1775即：P'2 :二 X £5 = 0.17756、(1)設X N(0,1)求 Z0.01； 對工2 分布，求 丁晨岱)；(3)對 t0.05(13)； (4)對 F 分布，求 Fo.o5(15,10 )。解：(1)編程求解：>> norminv(0.99)ans = 2.3263即：z0.01 =2.3263(2)編程求解：>

6、> chi2inv(0.95,8)ans = 15.5073即：%。5(8 )=15.5073(3)編程求解：>> tinv(0.95,13)ans = 1.7709即：to.o5 (13 )=1.7709(4)編程求解：>> finv(0.95,15,10)ans = 2.8450即：F0.05(15,10 )=2.84507、分別生成6父2個和6父1個均勻分布U (0,1)的隨機數。解：編程求解：>> A=unifrnd(0,1,6,2),B=rand(6,1)A =0.9501 0.45650.2311 0.01850.6068 0.82140.

7、4860 0.44470.8913 0.61540.7621 0.7919B =0.41030.89360.05790.35290.81320.0099練習5.31 .設X U (1,11 ),求該均勻分布的均值和方差。解：編程求解：>> m,v=unifstat(1,11) m = 6v = 8.33332 .設X N(0,16 ),求該正態分布的均值、標準差與方差。解：編程求解：>> x=normrnd(0,16,5,5);>> s=std(x),m,v=normstat(0,16)s = 21.5058 9.9310 14.5103 19.2052 1

8、7.4124m = 0v = 2563.生成6列服從標準正態分布的隨機數，每列 200個數，每列中，標準差的均值都為1.解：編程如下：>> x=normrnd(0,1,200,6)x =1.08840.06572.46812.1338-0.3558-0.71920.5006-0.0123-0.6692-0.0029-0.3277-0.21992.7718-0.07700.2599-0.08950.08310.5750-0.1603-1.5586-0.3723-0.25500.43340.17010.42951.70261.3186-0.8742-1.2230-0.4958-1.96

9、68-0.4690-0.65310.4229-2.73591.2027-0.54600.09460.0622-0.1334-0.5350-0.1121-1.88840.2871-0.73590.53962.20900.5628-0.10800.9194-0.17930.8752-0.6057-0.0307-1.31610.51011.0847-1.25081.4352-1.3228-0.67260.24540.13690.86831.1948-1.0830-0.9024-1.4005-0.0156-0.80480.74310.1575-0.15480.9696-0.9385-0.7527-0.

10、1214-1.46640.94721.5937-1.4781-0.7458-0.03120.67361.5504-1.43790.3619-0.3097-1.00300.36250.4290-1.53420.4778-1.5219-1.03810.6065-0.5608-0.07470.32170.82650.62860.07420.17930.0815-1.8776-0.61300.86781.0805-0.7715-0.84320.68050.95972.0718-0.6624-0.9434-0.56460.23341.9730-0.59440.4754-1.4076-0.02821.23

11、950.29500.58631.2443-1.9061-1.24370.1257-0.39271.52560.0296-0.06530.73300.17970.57592.14320.69170.67210.0596-0.6051-1.1414-0.7460-0.68560.20610.1491-1.03690.0611-1.5315-0.0431-0.00811.5959-0.29530.0123-0.2132-1.54770.0200-0.77731.4561-0.16810.4925-1.0718-0.55841.55031.8025-0.6873-0.3233-0.22111.8861

12、1.0550-1.3336-0.99070.8222-1.6758-0.2200-0.16670.3873-0.0498-1.4143-1.6981-1.41440.3145-0.02280.71931.1437-0.1085-0.30281.41960.1106-0.28310.9790-0.3008-0.56960.32730.8128-1.42500.4926-1.3683-0.12150.4757-1.00910.46151.25790.7377-0.39020.3988-1.00461.09150.5941-0.4043-0.8443-0.07280.2830-1.04430.454

13、50.8568-1.73781.31480.2898-2.8428-1.41783.3437-0.44950.9783-0.24730.9968-0.91990.6265-1.54791.7221-0.21890.0765-1.44811.2796-0.0958-0.41230.8987-1.8667-1.4813-0.74120.90770.5651-0.6422-0.61360.09731.03412.36960.7399-0.18041.1694-0.2263-0.61820.51980.22010.7179-0.5750-0.31720.42380.41051.31280.3014-0

14、.26480.63400.89491.05260.62921.54890.00470.0390-0.23750.4288-1.1080-0.0442-0.0394-0.0164-0.12791.2951-0.4470-0.0297-0.50540.44951.0195-0.1861-0.7260-0.3821-1.1578-0.52231.74840.13070.3540-0.55390.7104-1.05510.9875-0.6576-0.50680.93240.7282-0.0478-0.4201-0.7593-2.1037-1.31580.8669-0.4990-0.3337-0.595

15、2-0.6647-0.30152.43161.6275-0.85780.81241.4501-2.59960.1102-0.2390-0.77160.0695-0.32980.78010.0264-0.96171.4643-1.83372.70190.60290.9703-0.15271.09181.8274-1.63490.9428-0.00531.6830-0.21680.6541-0.5363-1.02391.40951.55511.4199-1.54480.5472-0.06781.7579-1.05020.6269-0.37511.49260.08180.88500.09672.22

16、150.2077-0.4552-1.76701.14090.2516-1.2924-0.7656-0.4964-1.78130.40322.24721.1703-0.10641.2353-0.66040.19100.3105-1.17890.33880.04091.3514-0.69361.9916-0.56791.03350.74852.13640.0110-0.11931.1773-1.40481.23080.1668-1.1056-0.27270.2257-1.03060.3049-1.70521.90801.44450.7576-0.64341.07780.2765-0.1654-0.

17、0744-0.84840.17080.76520.3945-0.7324-0.36251.85951.3448-1.3196-0.0986-0.9907-0.7770-0.03601.9363-0.50920.17640.8943-0.25012.59150.74130.5551-1.83790.4482-0.4187-0.69130.8120-1.9576-1.50230.88921.4149-1.5765-0.1428-0.76050.81921.0733-0.8474-0.6101-0.0999-2.4439-0.2346-0.10470.16610.3767-0.8001-0.6594

18、-1.63161.5473-0.04821.27280.4932-0.1148-0.31790.1698-2.9772-0.59181.23760.3001-0.79630.80401.00702.24961.2960-0.58400.6908-1.72400.8844-0.0163-0.2782-3.0737-0.04200.1741-0.40590.73580.21711.55100.3240-0.4841-2.6192-0.64090.6307-0.40740.5065-0.7316-0.96861.4443-0.54851.4281-1.0286-2.13190.5553-1.1590

19、0.2296-1.35320.0994-1.81040.75950.68630.35530.9040-0.1164-0.0523-0.57200.73040.52130.54170.6892-0.08621.15800.5145-0.6160-0.46501.8833-1.1897-0.30801.69581.34582.43040.3254-0.75410.0032-0.67630.97492.0205-0.09520.9473-1.40613.4128-2.37790.79730.0312-0.18261.6241-0.3947-1.09230.0310-0.6138-0.06630.13

20、961.2059-0.32570.5407-1.73130.9050-1.81660.3078-2.01220.68390.47881.4582-2.7892-0.48641.5677-0.5901-0.4478-1.11800.2624-0.33100.2333-0.26110.38681.81330.11920.77670.64641.51710.05300.15080.3323-0.3327-1.12941.0073-0.4861-0.2830-0.44652.09630.19700.30340.24451.65010.00520.38881.6969-0.81710.71830.666

21、4-0.2061-0.65250.7260-0.49120.1535-0.34640.0495-0.05680.79250.86750.1338-0.26400.92600.42250.60340.3608-1.0062-0.64430.3508-0.0749-0.0584-0.08041.3065-0.9055-0.79580.7867-1.10870.74931.19910.7167-0.6222-0.13712.1442-1.7920-2.5773-0.00731.42990.6365-1.35281.2132-2.0863-2.8148-2.08491.57810.4570-0.060

22、50.3861-0.14950.0337-0.03090.3912-0.3925-0.86100.5775-0.01570.31632.07300.6095-1.23080.7531-0.55431.9848-0.32330.64362.6416-0.1670-1.1636-0.64501.46811.0195-0.9044-0.58180.10000.3133-0.50240.9344-1.22330.36580.47831.59240.20961.22860.3032-0.54891.7747-0.97320.7548-0.2495-0.73010.5787-0.4705-1.1477-0

23、.9482-0.7076-1.1436-1.9558-0.9253-1.48450.6132-0.5938-1.41320.52200.0015-0.96611.7605-0.2623-0.59181.60110.6373-0.03970.08881.24280.51890.7326-1.5689-2.12192.5956-1.5489-1.49280.55570.7002-1.8364-0.6755-0.3868-0.0867-0.81330.9738-1.54202.78680.2751-0.0126-0.5390-0.0705-0.2354-0.01680.8262-0.34591.30

24、79-0.03371.19940.2717-0.97930.9863-0.48260.3025-0.0332-0.9141-0.10430.6433-1.7524-0.6566-0.5414-1.95140.12782.9199-0.9266-0.0522-0.0674-0.31740.0625-1.24860.92240.77322.25600.58830.37160.15710.0410-0.31990.21240.8290-0.10400.78870.4136-1.12060.1884-1.6749-0.6968-0.57710.59721.46060.2847-1.9223-0.386

25、80.52761.92430.4091-1.5296-0.43670.01611.67170.71411.1326-2.37020.04501.36920.8001-2.31230.5700-2.03212.41660.41690.88381.3807-0.91771.2202-0.30990.0687-0.22421.39071.7010-0.36480.18760.29420.2970-0.4539-0.0248-0.63280.94770.4726-0.5210-0.6283-0.7698-0.2226-0.52571.7827-0.15541.0232-0.83750.1806-1.1

26、1560.2606-0.0985-0.4891-0.09180.8971-1.59231.51070.9972-0.40240.6301-1.23051.17480.31700.43451.39970.38450.12540.48510.8035-0.02570.2806-0.60990.80741.64550.5809-0.37991.8809-0.4137-1.0653-0.45421.7786-0.2424-0.79971.1181-2.00911.0088-0.9381-1.1651-0.5607-0.70030.24322.0494-0.91670.75661.7089-0.3501

27、0.14950.60200.37601.1642-0.64481.81682.60780.01790.9098-1.0235-1.4225-1.6401-0.0532-1.61040.15421.7016-0.75590.2771-0.01471.2388-0.2023-0.49420.1575-0.6745-0.51670.68361.48870.17270.37830.20391.2073-0.7807-0.62160.35410.17871.30740.11080.53100.8095-0.2463-0.60280.46211.69352.13451.9288-0.1457-0.9934

28、1.28850.45730.35440.3961-1.16901.1889-0.95611.06710.2317-0.8614-0.02202.38801.1957-0.12761.28802.43190.61832.2655-0.40560.1880-0.0135-0.84051.86592.3011-0.6034-0.4533-1.33330.28050.0819-0.27011.8336-0.6201-0.55630.82041.60800.5028-0.9653-1.92540.75561.2278-0.3807-0.1192-0.3970-1.4255-0.9119-0.0636-1

29、.2996-0.00190.08980.26801.37170.6453-0.7240-0.43260.2640-0.36260.2456-1.7713-0.5650-0.19480.2922-0.98410.11880.05960.62170.98540.78181.36500.3847-0.7602-1.33550.46860.40400.8670-0.0702-1.6909-0.1231-1.3649-0.3254-0.4289-0.57831.1037-1.10280.2737-0.37390.97180.46931.4625-2.75322.6467-0.2943-1.14351.2

30、9970.23620.2520-0.0538-1.71792.04771.6348-1.0977-0.85810.4725-0.05390.0078-0.70282.41521.1354-2.08000.21790.74680.8073-0.4021-0.2979-0.80252.17320.5396-1.02750.91411.1543-0.45680.5724-0.69181.2945-0.13601.04610.19390.81501.08410.01491.31422.12690.8895-1.07890.62940.21870.3224-0.6558-1.5917-0.5799-0.

31、75611.7132-0.4765-1.1424-0.3220-1.87570.2388-2.07880.07620.9490-0.70380.9175-1.20330.1129-0.1051-0.4046-0.7443-0.5469-0.3082-1.08651.4170-0.38430.3713-0.6051-1.5379-1.55830.70790.48201.43730.0253-0.62030.63740.36790.44380.45990.2615-0.21480204060x-40-20-0.4046-0.60280.38110.66070.74351.0820-0.4033-0

32、.85211.10231.1244-0.1620-0.17100.08410.65510.85640.97940.4357-0.8103-0.43531.4702-1.1785-1.31640.8613-0.0057-0.5626-0.81040.4020-0.02320.0641-1.16290.8781-1.2762-0.58420.1345-1.7273-0.6471-0.81461.7223-0.97952.40810.71601.5723-0.25840.10190.11510.90170.03660.13020.4933-0.80200.06850.0762-0.1849-0.80

33、43-0.8027-1.2508-0.52990.3617-0.8147-0.1024-0.00831.23770.5411-2.05870.9900-0.66760.62761.52820.6817-2.3320-1.7817-0.70570.15441.77690.5386-0.3709-0.0440-0.22012.58070.6312-0.51001.28570.8902-0.4506-1.30620.0833-1.32210.5570-0.45610.14081.02352.1400-0.6107-0.1802-1.90371.58140.77781.2635-0.5653-0.03

34、57-0.3921-0.2971-0.8339-1.75060.08621.9344-1.1070-0.3513-0.5867-0.01440.69151.30561.75750.87254、首先生成正態分布N (0,16)的容量為300的隨機數的樣本，然后畫正態分布N (0,16)的直方圖。解：編程求解：>> x=normrnd(0,16,300,1);>> hist(x,7)圖像：1009080706050403020100練習5.51 .泥廠用自動包裝機包裝水泥，每袋額定重量是50 kg,某日開工后隨機抽查了9袋，稱得重量如下：49.6 49.3 50.1 50.

35、0 49.2 49.9 49.8 51.0 50.2.設每袋重量服從正態分布，問包裝機工作是否正常？(取顯著性水平a =0.05.)解：假設檢驗：H0:=50,H1:;50編程如下：>> x=49.6 49.3 50.1 50.0 49.2 49.9 49.8 51.0 50.2;>> h,sig,ci=ttest(x,50) h =0sig = 0.5911 ci = 49.4878 50.3122檢驗結果為：布爾值h=0說明表示在顯著性水平為0.05下接受原假設H0,說明包裝機工作正常。置信水平為0.95的置信區間為(49.4878,50.3122 ),它包含50,

36、因此接受原假設。sig =0.5911 A0.05,也說明能接受“包裝機正常工作”的假設。2 .某工廠生產的某種型號的電池，其壽命(以小時計)長期以來服從方差為5000的正態分布，現有一批這種電池，從它的生產情況來看，壽命的波動性有所改變.現隨機取26只電池, 測出其壽命的樣本方差s = 9200.問根據這一數據能否推斷這批電池的壽命的波動性較以往的有顯著的變化？(取顯著性水平a = 0.05.)解：假設檢驗：H o： sig =5000,Hsig =5000編程如下：建立 M文件，命名為：Untitledsigma0=5000; % 總體原始方差sigma1=9200; % 樣本方差alph

37、a=0.05; %顯著性水平n=26; %樣本容量chi2stat=(n-1)*sigma1/sigma0; %卡方檢驗統計量criticalValue1 =chi2inv(alpha/2,n-1); %臨界值criticalValue2=chi2inv(1-alpha/2,n-1); %臨界值if (chi2stat>criticalValue1&&chi2stat<criticalValue2) %判斷并顯示結論disp('接受原假設，認為方差沒有改變)elsedisp('拒絕原假設，認為方差發生了改變 )end運行M文件，得結果：拒絕原假設，認

38、為方差發生了改變3、某地某年高考后隨機抽得15名男生、12名女生的數學考試成績如下：男生：119 118 117 123 121 113 109 127 116 116 112 114 125 114 110女生：116 110 117 121 113 106 113 108 118 124 118 104從這27名學生的成績能說明這個地區男、女生的數學考試成績不相上下嗎？(顯著性水平口 =0.05.)解：假設：H 0 :1 - - 2驗證數學成績服從正態分布編程如下：>> x1=119 118 117 123 121 113 109 127 116 116 112 114 125

39、 114 110;>> x2=116 110 117 121 113 106 113 108 118 124 118 104;>> subplot(1,2,1);normplot(x1);subplot(1,2,2);normplot(x2)圖像：由于正太概率圖都顯示出直線形態，因此數據 檢驗編程如下：x1和數據x2都可以認為如從正態分布> > x1=119 118 117 123 121 113 109 127 116 116 112 114 125 114 110;> > x2=116 110 117 121 113 106 113 108

40、118 124 118 104;> > pt,sigt=ttest2(x1,x2)pt =0sigt = 0.1945可見，男、女生數學成績不相上下，沒有顯著差異，接受假設。4、下面列出84個伊特拉斯坎男子頭顱的最大寬度(單位：mm： 141 148 132 138 154142 150146155158150140147148144150149145149158143141 144144126140144142141140145135147146141136140146 142137148154137139143140131143141149148135148152 143144

41、141143147146150132142142143153149146149138 142149142137134144146147140142140137152145請檢驗上述頭顱的最大寬度數據是否來自正態總體？(顯著性水平a = 0.05.) 解：編程：x=141 148 132 138 154 142 150 146 155 158 150 140 147 148 144 150 149145 149 158 143 141 144 144 126 140 144 142 141 140 145 135 147 146 141 136 140146 142 137 148 154 13

42、7 139 143 140 131 143 141 149 148 135 148 152 143 144 141143 147 146 150 132 142 142 143 153 149 146 149 138 142 149 142 137 134 144 146147 140 142 140 137 152 145;>> normplot(x)圖像：0.9970.003,十士牛潭豐=a4-十-4-'十七 丁1.Normal Probability Plot0.950.900.750.250.100.051301351501550.990.980.020.01b 0

43、.50140145Data壽命t/h0,100】(100,200】(200,300>300燈泡數121784358由于正太概率圖都顯示出直線形態，因此數據x1和數據x2都可以認為如從正態分布5、在一批燈泡中抽取 300只做壽命試驗，獲得的數據見下表對于給定的顯著性水平 a =0.05,問這批燈泡的壽命是否服從指數分布解：編程：>> t=0:100:300;>>>>h=121 78 43 58;pi=0.005*exp(-t*0.005)f (t )=pi = 0.0050 0.0030 0.0018 0.0011>>>>t=40

44、0 500 600 700;sum(0.005*exp(-t*0.005)ans = 0.0015>> n=300;一 -L 一一0.005e ,t >0,> > sum(h-n*pi)，2/(n*pi)ans = 8.2806e+003> > syms x> > ff=(x)(chi2pdf(x,4);> > p=quadl(ff,ans,10000)p = 0由于p =0 <0.05 = a ,所以顯著性水平a =0.05下，這批燈泡的壽命不如從指數分布。6.謀電話站在一個小時內接到電話用戶的呼叫次數按每分鐘記錄如下表

45、呼叫次 數0123456>7頻數81617106210問在顯著性說平a =0.05時，在一個小時內接到電話用戶的呼叫次數能否看作來自泊松分 布？解：編程求解：>> i=0:1:7;> > ni=8 16 17 10 6 2 1 0;> > sum(i.*ni)./60)ans =2> > pi=(2.Ai)./factorial(i).*exp(-2)pi = Columns 1 through 60.1353 0.2707 0.2707 0.1804 0.0902 0.0361Columns 7 through 8 0.0120 0.00

46、34> > i=8 9 10 11 12 13 14 15;> > sum(2.Ai)./factorial(i).*exp(-2)ans = 0.0011> > n=60;> > sum(ni-n*pi).A2)./(n*pi)ans = 0.4937> > syms x> > ff=(x)(chi2pdf(x,8);> > p=quadl(ff,0.4937,10) p = 0.7348由于p =0.7348 >0.05 = a ,所以顯著性水平 a = 0.05下，可以認為“在一小時接到電話 用戶的

47、呼叫次數如從泊松分布。練習5.61.某地區車禍次數 y (千次)與汽車擁有量 x (萬輛)的11年統計數據如下表年度1234567891011汽車 擁啟 量/ 萬輛325373411411462490529577641692743車禍 次數 /千次166153177201216208227238268268274（1）作y和x的散點圖；（2）如果從（1）中的散點圖大致可以看出y對x是線性的，試求線性回歸方程；（3）驗證回歸方程的顯著性（顯著性水平a =0.05）; （4）假設擁有800萬輛汽車，求車禍次數置信水平為0.95的預測區間.解：（1）編程如下：>> x=352 373 4

48、11 411 462 490 529 577 641 692 743;>> y=166 153 177 201 216 208 227 238 268 268 274;>> plot（x,y,'*'） 圖像：280260240220200180160140350400450500550600650700750>> X=ones(11,1),x'>> b,bint,r,rint,s=regress(y',X,0.05) b = 55.85270.3120bint = 23.0712 88.63420.2506 0.3

49、734r = 0.3364-19.2149-7.06957.571716.0204-0.71456.11892.144712.1791-3.7311-13.6412rint =-22.7953 23.4680-37.2950 -1.1347-30.9881 16.8492-16.7413 31.8846-5.7896 37.8305-26.2054 24.7763-18.9682 31.2061-23.0862 27.3756-10.0894 34.4475-26.4931 19.0310-31.5901 4.3077s = 0.9362 132.0614 0.0000 124.9076AA因此 b1 =55.8527,b2 =0.3120, b1 的置信水平為 0.9