Solution Manual for Mathematical Statistics with Applications, 8th Edition

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SOLUTIONSMANUALJOHNE.FREUND’SMATHEMATICALSTATISTICSWITHAPPLICATIONSEIGHTHEDITIONIrwin MillerMaryless Miller

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iiiTable of ContentsChapter 1..............................................................................................................................1Chapter 2..............................................................................................................................8Chapter 3............................................................................................................................23Chapter 4.............................................................................................................................46Chapter 5.............................................................................................................................61Chapter 6.............................................................................................................................80Chapter 7............................................................................................................................95Chapter 8..........................................................................................................................112Chapter 9...........................................................................................................................130Chapter 10.......................................................................................................................139Chapter 11.......................................................................................................................162Chapter 12.......................................................................................................................171Chapter 13.......................................................................................................................182Chapter 14.......................................................................................................................199Chapter 15.......................................................................................................................226Chapter 16.......................................................................................................................245Appendix A......................................................................................................................258

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1Chapter 11.1(a)121niin(b)0 1 2 30 1 2 30 1 20 1131.211221211nniin inn n1.3(a)3004n3203n3014n3212n3023n3221n3032n3302n3104n3311n3113n3122n3131n(b)443...21321.412212311nnijnn n n

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2Mathematical Statistics, 8E1.5(b)6, 20, and 70“2 out of 3”m= 21222(12)611   “3 out of 5”m= 323422(136)20222“4 out of 7”m= 4345622(141020)703333      1.6(a)10101010!0(7.92665)(3.678797)(7.92665)(454,002.49)3,598,719eπ% error3.62883.5987 1000.83%3.628812121212!24(8.683215)(4.41455)475,683,224eπ4.78004.7568% error1000.69%4.7900(b)52133952525213393939521045252!1313! 39!133926781344639 billion19.51313319.53eeeπππππ1.7Using Stirling’s formula in22 !!!nnnnn yields22222241222nnnnnnnnnneneπππππ1.8rnand312= 1,7281.915317and2155rnr 1.10Substituternforrinto result of 1.911514and6532rnnrrnrn

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Chapter 131.11(b)Seventh row is 1, 6, 15, 20, 15, 6, 1Eighth row is 1, 7, 21, 35, 35, 21, 7, 166542232456()61520156xyxx yx yx yx yxyy7765243342567()7213535217xyxx yx yx yx yx yxyy1.14(a)Setx= 1 andy= 1(b)Setx= 1 andy=1(c)Setx= 1 andy=a11.19(a)113515222224384        and( 3)( 4)( 5)106 (b)12152 14231111111312 124224222411351264832 1286451251257122.23512         11422.2305121.20(a)( 1)( 2)...()( 1)!rrr(b)()(1)...(1)(1)...(1)( 1)!!1(1)...(1)( 1)( 1)!rrrnnnnrn nnrrrrnrnrnnrr  1.218!8 7 6 5 45602! 3! 3!2 61.223239!9 8 7 6 5 423( 4)8 9 6423,224,3203! 2! 3!12  

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4Mathematical Statistics, 8E1.24Note: If there are 0 turn-ons the first night, 6 turn-ons in four nights can only occur if there are 2turn-ons on each of the subsequent three nights. Thus, we need to show only that part of the treefollowing this event.

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Chapter 151.251.26(a)(b)1.27(a)5(b)41.281.291.30(a)6 530;(b)6 636;1.31(a)6;(b)6 530;(c)5 420first one fixed;(d)6302056

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6Mathematical Statistics, 8E1.32(a)4 5 240;(b)5 6 3901.33(a)5 420;(b)5 4 3601.3415314,348,9071.3515 141052 11.36(a)10 9 8 75040;(b)5040210241.37(a)14 1391;2 1(b)14 13 123643 2 11.386!7201.396!720902! 2! 2!81.405!120and1202 4!721.417!50401.42(a)5!120;(b)5!602!1.4310!362880050,4003! 3! 2!72and8!4032033603! 2!121.4410!36288001,2605! 4!120 241.458!403202803! 4!6 241.46(a)2077,5207 ;(b)20184,75510 (c)2020202011401902011351171819201.47(a)7212 ;(b)462 ;(c)3 412

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Chapter 171.4837373 211 7637702231         1.494736 35 3630231          1.50131313131287 286 286 788,211,173,2565332          1.517!50404203! 2!121.5210359,0491.535515,6251.54126117176,188121251.5512111462661.5614311612014141.5721111rnnrnnn1104512rnn

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139Chapter 1010.1()iiiiiiEa xa E xaaμμ 11niia10.211221212ˆˆ[],1E kkkkkkθθθθθ10.3(21)!( )( )( )( )!!mmxxmh xfxdxfxfxdxmm(1/2)(1/2)(21)!( )1!!(21)!111!!22mmxxmmmh xdxdxmmmxxmm mθθθθ 11( )6(22h xxxθθ   (1/2)(1/2)11( )622E xxxx dxθθθθ   letu=12xθ1016(1)2uuu duθθ10.42//6()18xxh xeeθθ2//02/3 /006[]1665biased6xxxxE xx eedxx edxx edxθθθθθθθθUse gamma integrals.

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140Mathematical Statistics, 8E10.5221222111()(11niiiniExExnnnnnμμσσσ10.6()E xμ2var()xnσ2222()E xnσμμasn 10.71111(1)(1)22222xnEE xnnnnnnθθ whenEnθ , so is asymptotically unbiased10.8111211()()11()(1)()()()nyxyynyn ygyn eedxn een eδδδδδ1()11110()let1()n ynuE ynyedyuynuedunδδδδδThe unbiased estimate is11Yn1()asE Ynδ 10.911111111()()nnnyngyndxyβββββ11111110111100()()()(1)1nnnnnydynE YyydyudubuudunuudunββββββββββββUnbiased estimate is1(1)nY10.102211222211111()nniiiinniixEE xnnnnσμσσ 

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Chapter 1014110.1122211( )()1(1)(1) (1)(1)biasedxxEnE xE xnnnnnnnnnθθθθθθθθ10.12(a)1nvalues beforenyin11nynways.11()nnynfyknfor,,nynk(b)111()1(1)see Exercise 1.15 or Theorem 1.11, respectively1nnnkknnnynynyyknnnE Yykkknnnnnn kn 11(1)11QED1nnnn kEYknnn10.13222ˆˆˆˆ()var( )( )var( )EEθθθθθ22()since var( )0Eθθθ10.141( ;)(1)xxfxθθθ( )E xθ2()E xθln( ;)ln(1)ln(1)fxxxθθθln( ;)11(1)fxxxxθθθθθθθ2222ln( ;)11()(1)(1)fxEE xθθθθθθθ1(1)varxn Enuθθwhenxis binomial random variable.xnis minimum variance estimatorxnEnnθθ unbiased

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142Mathematical Statistics, 8E10.15( ;)!xefxxλλλμλ2σλvar()xnλ()unbiasedE xλlnlnln!fxxλλln1fxλλ22222ln()2( )1211fE xEE xλλλλλλλλλ1var()xnEnλxis minimum variance unbiased estimator10.1612ˆˆvar()3var()θθ11221212ˆˆ()1E aaaaaaθθθθθ22122ˆˆvarvar()var()iaaθθ222222212222112ˆˆˆvar3var()var()(3) var()ˆ[3(1) ]var()iaaaaaaθθθθ21111262(1)( 1)1382044aaaaaa10.17/1( ;)xfxeθθθ( )E xθ22()2E xθ22σθ()unbiasedE xθ2var()xnθlnlnxfθθ 22ln1fxxθθθθθ 2242ln11()fEE xθθθθ21var()xxnEnθis minimum variance unbiased estimator10.182222(),(), var()12(2)(1)nnnnnnE YE YYnnnnβββlet1nnBYn1()unbiased1nnE Bnnββ

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Chapter 1014322222(1)var()(2)(2)(1)nnBn nnnnββ22211var()1(2)ln()Bnn nfXnnEββββso the Cramèr-Rao inequality is not satisfied.10.19(a)ln( )1( )( )fxfxfxθθ( )ln( )( )fxfxfxθθln( )( )0fxfx dxθ(b)22ln( )ln( )ln( )( )( )fxfxfxfxfxθθθ222ln( )ln( )( )( )fxfxfx dxfx dxθθ 22ln( )ln( )fxfxEEθθ 10.20ln( )1fxxμμσσfrom Example 10.5222ln( )1fxμσ 222111ln( )nfxnnEσσμ10.21(a)12[(1)](1)E wxw xwwμμμ(b)22221212var[(1)](1)wxw xwwnnσσ221222(1)( 1)0dwwdwnnσσ222122wσσσ222212wσσσ10.22222212var1(1)wwnnσσ12w222212121var()444nnnσσσσ

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144Mathematical Statistics, 8E2222222112222212122222221221122222222121212var 2()nnnnσσσσσσσσσ σσσσ σσσσσσσ2212222221212222222121212221222212()4efficiency1()()44()nnnnσσσσσ σσσσσσσσ σσσ10.23222212var(1)wwnnσσ221222(1)0dwwdwnnσσ121wwnn112nwnn10.24For12w22212121111var444nnnnσσσFor112nwnn222212121122221221212var()()nnnnnnnnnnnnnnσσσσ Efficiency =212122212124()114n nnnnnnnσσ10.25222212321113var4164168xxxσσσσ 2var()3xσEfficiency =2283398σσ

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