{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 } 1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 43 "Probability Distribution \+ Function and Shape" }}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 257 18 "The F Distribution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The F distribut ion was originally derived as the ratio of two independent variables w ith " }{XPPEDIT 18 0 "chi^2;" "6#*$%$chiG\"\"#" }{TEXT -1 320 " distri butions, each divided by its respective degrees of freedom, which is w hy the parameters of the F distribution are called degrees of freedom \+ as well. The F distribution often arises in the comparison of the var iances of two normal populations. The F distribution is also known as the variance-ratio distribution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "A \+ random variable has a F(" }{XPPEDIT 18 0 "nu[1],nu[2];" "6$&%#nuG6#\" \"\"&F$6#\"\"#" }{TEXT -1 65 ") distribution if and only if its probab ility density is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 18 " " }{XPPEDIT 18 0 "Gamma((nu[1]+n u[2])/2)*(nu[1]/nu[2])^(nu[1]/2)*x^(nu[1]/2-1)*(1+nu[1]*x/nu[2])^(-(nu [1]+nu[2])/2)/(Gamma(nu[1]/2)*Gamma(nu[2]/2));" "6#*&**-%&GammaG6#*&,& &%#nuG6#\"\"\"\"\"\"&F+6#\"\"#F.F.\"\"#!\"\"F.)*&&F+6#\"\"\"F.&F+6#\" \"#F3*&&F+6#\"\"\"F.\"\"#F3F.)%\"xG,&*&&F+6#\"\"\"F.\"\"#F3F.\"\"\"F3F .),&\"\"\"F.*&*&&F+6#\"\"\"F.FBF.F.&F+6#\"\"#F3F.,$*&,&&F+6#\"\"\"F.&F +6#\"\"#F.F.\"\"#F3F3F.F.*&-F&6#*&&F+6#\"\"\"F.\"\"#F3F.-F&6#*&&F+6#\" \"#F.\"\"#F3F.F3" }{TEXT -1 5 " for " }{TEXT 259 2 "x " }{TEXT -1 4 "> 0 " }}{PARA 0 "" 0 "" {TEXT 272 1 "f" }{TEXT -1 3 " ( " }{TEXT 270 0 "" }{TEXT -1 0 "" }{TEXT 271 2 "x " }{TEXT -1 4 ") = " }{TEXT 269 0 " " }}{PARA 0 "" 0 "" {TEXT -1 60 " 0 \+ elsewhere" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "nu[1],nu[2];" "6$&%#nuG6#\"\" \"&F$6#\"\"#" }{TEXT -1 30 " are the degrees of freedom. " }{XPPEDIT 18 0 "nu[1], nu[2]" "6$&%#nuG6#\"\"\"&F$6#\"\"#" }{TEXT -1 23 " are po sitive integers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The following code will draw the density function for the F(" }{XPPEDIT 18 0 "nu[1],nu[2];" "6$&%#nuG6#\"\"\"&F$6#\"\"#" } {TEXT -1 19 ") distribution for " }{XPPEDIT 18 0 "nu[1] = 7;" "6#/&%#n uG6#\"\"\"\"\"(" }{TEXT -1 4 "and " }{XPPEDIT 18 0 "nu[2]" "6#&%#nuG6# \"\"#" }{TEXT -1 52 "=10, but you are encouraged to try other values o f " }{XPPEDIT 18 0 "nu[1],nu[2];" "6$&%#nuG6#\"\"\"&F$6#\"\"#" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nu[1]:=7; nu[2]:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6 #\"\"\"\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6#\"\"#\"#5" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:=x->FPDF(nu[1],nu[2],x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&ar rowGF(-%%FPDFG6%&%#nuG6#\"\"\"&F06#\"\"#9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(f(x),x=0..10, title=\"PDF for F(7,10)\") ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7do7$\"\"!F(7$$\"1LL$3FWYs#!#<$ \"1VA&pHT'p_!#=7$$\"1mmmT&)G\\aF,$\"1d%)>a%zia#F,7$$\"1++]7G$R<)F,$\"1 avw^Sx5gF,7$$\"1LLL3x&)*3\"!#;$\"1'4'3eZ#*f5F=7$$\"1mmTN@Ki8F=$\"1U%G: v.[f\"F=7$$\"1++]ilyM;F=$\"1.l$)GbQs@F=7$$\"1LLe*)4D2>F=$\"1a\"z-k(ykF F=7$$\"1nmm;arz@F=$\"1V0Lk,+]LF=7$$\"1+D\"yD&y;CF=$\"1d!Rj\"\\LSQF=7$$ \"1L$e*)4bQl#F=$\"1Ri)zrkYI%F=7$$\"1mT5S\\#4*GF=$\"1)oSq%>LPZF=7$$\"1+ +D\"y%*z7$F=$\"1>Zt)[(eM^F=7$$\"1m;ajW8-OF=$\"1;LF!z,`\"eF=7$$\"1LL$e9 ui2%F=$\"1ViqWWXUjF=7$$\"1n;H2Q\\4YF=$\"10*e&QwwhnF=7$$\"1++voMrU^F=$ \"1a*o$Q]TT8(F=7$$\"1+v$4'HaUfF=$\"1lea!*)e#[rF=7$$\"1nmm\"z_\"4iF=$\"1=*[ bMqp8(F=7$$\"1nmmm6m#G(F=$\"1._.#y\"f*)oF=7$$\"1ommT&phN)F=$\"1P&e-w3& HkF=7$$\"1,+v=ddC%*F=$\"1!oykb(epeF=7$$\"1LLe*=)H\\5!#:$\"1afMOWLy_F=7 $$\"1++v=JN[6F\\s$\"1i#[Y)zeRZF=7$$\"1nm\"z/3uC\"F\\s$\"1^q'peR*HUF=7$ $\"1LLe*ot*\\8F\\s$\"1e<$)Q,`VPF=7$$\"1++DJ$RDX\"F\\s$\"1.]]]r^.LF=7$$ \"1LLekGhe:F\\s$\"1Q,Owld(*GF=7$$\"1nm\"zR'ok;F\\s$\"1&fN/x\"F\\s$\"1i6wb9&eA#F=7$$\"1++D1J:w=F\\s$\"1C@FI#4=&>F=7$$\"1M LL3En$4#F\\s$\"1T\\9Q9c$\\\"F=7$$\"1nm;/RE&G#F\\s$\"1;G$f*o7&=\"F=7$$ \"1+++D.&4]#F\\s$\"1g65c`\\*=*F,7$$\"1+++vB_;c+$F,7$$\"1++D\"o 7Tv$F\\s$\"1x7%4d6DT#F,7$$\"1LLL$Q*o]RF\\s$\"1q]%=#3O(*>F,7$$\"1,+D\"= lj;%F\\s$\"1dh*)=ICL;F,7$$\"1++vV&RLULWmF/7$$\"1++v=> Y2aF\\s$\"1ufON.14dF/7$$\"1nm;zXu9cF\\s$\"1_S79%*el[F/7$$\"1+++]y))GeF \\s$\"1M0#**[8?9%F/7$$\"1++]i_QQgF\\s$\"1,E$)G:?_NF/7$$\"1,+D\"y%3TiF \\s$\"1*pIKxbB2$F/7$$\"1++]P![hY'F\\s$\"1KZZ#HP_i#F/7$$\"1LLL$Qx$omF\\ s$\"1v3T8f$oG#F/7$$\"1+++v.I%)oF\\s$\"1_A6([v+)>F/7$$\"1mm\"zpe*zqF\\s $\"1]2X=\"GFu\"F/7$$\"1,++D\\'QH(F\\s$\"1g)QS8g,_\"F/7$$\"1LLe9S8&\\(F \\s$\"1)49a#zQS8F/7$$\"1,+D1#=bq(F\\s$\"13BH%=G$y6F/7$$\"1LLL3s?6zF\\s $\"1l3d?u]T5F/7$$\"1++DJXaE\")F\\s$\"1?R!3'yYw\"*!#>7$$\"1ommm*RRL)F\\ s$\"1kfd&=8F9)F]_l7$$\"1om;a<.Y&)F\\s$\"1)eE$3S&HA(F]_l7$$\"1NLe9tOc() F\\s$\"1vwu$zd!GkF]_l7$$\"1,++]Qk\\*)F\\s$\"18fOM%Ghy&F]_l7$$\"1NL$3dg 6<*F\\s$\"1`Y=3\"*3S^F]_l7$$\"1ommmxGp$*F\\s$\"14bkb:ZKYF]_l7$$\"1++D \"oK0e*F\\s$\"1^FQR&3V:%F]_l7$$\"1,+v=5s#y*F\\s$\"1Vv!Rfj'\\PF]_l7$$\" #5F($\"1;aN7&)*\\O$F]_l-%'COLOURG6&%$RGBG$Fhal!\"\"F(F(-%+AXESLABELSG6 $Q\"x6\"%!G-%&TITLEG6#Q0PDF~for~F(7,10)Febl-%%VIEWG6$;F(Fgal%(DEFAULTG " 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "To see the effe ct of " }{XPPEDIT 18 0 "nu[1];" "6#&%#nuG6#\"\"\"" }{TEXT -1 23 " on t he shape of the F(" }{XPPEDIT 18 0 "nu[1],nu[2];" "6$&%#nuG6#\"\"\"&F$ 6#\"\"#" }{TEXT -1 95 ") distribution, the following animation will dr aw a series of probability density functions as " }{XPPEDIT 18 0 "nu[1 ];" "6#&%#nuG6#\"\"\"" }{TEXT -1 54 " varies from 1 to 21 by increment s of 1 while holding " }{XPPEDIT 18 0 "nu[2];" "6#&%#nuG6#\"\"#" } {TEXT -1 18 " constant at 10. 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We will denote the PDF by " }{TEXT 273 1 "f" }{TEXT -1 16 " and t he CDF by " }{TEXT 274 1 "F" }{TEXT -1 88 ". The CDF for the F(7,10) \+ appears below, but you are encouraged to try other values of " } {XPPEDIT 18 0 "nu[1]" "6#&%#nuG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "nu[2]" "6#&%#nuG6#\"\"#" }{TEXT -1 2 ". 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celF(7&FgelFbel7$$\"1++v$pU&G5F=$\"1.9ON+h([*F,7$FjelF(7&F^flFiel7$$\" 1LLLe0$=C\"F=$\"165*>RM6N\"F=7$FaflF(7&FeflF`fl7$$\"1LLLLA`c9F=$\"1q'p v'HA\"z\"F=7$FhflF(7&F\\glFgfl7$$\"1LLL3RBr;F=$\"1G**)[B88D#F=7$F_glF( 7&FcglF^gl7$$\"1++vV^\"\\)=F=$\"1`nHILB;FF=7$FfglF(7&FjglFegl7$$\"1mm; zjf)4#F=$\"1r3:F,gxJF=7$F]hlF(7&FahlF\\hl7$$\"1++]Piq'H#F=$\"1j_I5Jy%f $F=7$FdhlF(7&FhhlFchl7$$\"1LL$e4;[\\#F=$\"1m(Q+oej*RF=7$F[ilF(7&F_ilFj hl7$$\"1mm;zt%**p#F=$\"1B_b!)pS\"R%F=7$FbilF(7&FfilFail7$$\"1++]i'y]!H F=$\"1\"HuVwr?w%F=7$FiilF(7&F]jlFhil7$$\"1mm;HdATWkF=7$Fe[mF(7&Fi[mFd[m7$$\"1LLL3y_qXF=$\"1q!G6&R&pt'F=7$F\\\\mF(7&F `\\mF[\\m7$$\"1+++]1!>+&F=$\"1,+E#QYW'pF=7$Fc\\mF(7&Fg\\mFb\\m7$$\"1++ ++FZ=_F=$\"1F9#)*et=/(F=7$Fj\\mF(7&F^]mFi\\m7$$\"1+++]Z/NaF=$\"1AYZ6s( p4(F=7$Fa]mF(7&Fe]mF`]m7$$\"1******\\?vVcF=$\"1')z\"yJ]08(F=7$Fh]mF(7& F\\^mFg]m7$$\"1+++]$fC&eF=$\"1@$Rt*GZYrF=7$F_^mF(7&Fc^mF^^m7$$\"1mm\"H d&)>/'F=$\"1)*o9F`'o9(F=7$Ff^mF(7&Fj^mFe^m7$$\"1LL$ez6:B'F=$\"1^'egQf \\8(F=7$F]_mF(7&Fa_mF\\_m7$$\"1++D1o(oX'F=$\"14qs%3&>1rF=7$Fd_mF(7&Fh_ mFc_m7$$\"1mmm;=C#o'F=$\"1C())39&4jqF=7$F[`mF(7&F_`mFj_m7$$\"1mmmm#pS1 (F=$\"1&e\\:$HghpF=7$Fb`mF(7&Ff`mFa`m7$$\"1++]i`A3vF=$\"1?X'f%z@1oF=7$ Fi`mF(7&F]amFh`m7$$\"1mmmm(y8!zF=$\"1g$\\P,EBk'F=7$F`amF(7&FdamF_am7$$ \"1++]i.tK$)F=$\"1$*>S6I(4W'F=7$FgamF(7&F[bmFfam7$$\"1++](3zMu)F=$\"1C )[e%G(QB'F=7$F^bmF(7&FbbmF]bm7$$\"1nmm\"H_?<*F=$\"1HsJF=7$FghmF(7&F[imFfhm7$$\"1++DTO5T:F\\s$\" 1Ht?gh@hHF=7$F^imF(7&FbimF]im7$$\"1nmmT9C#e\"F\\s$\"1Gk6q_v8GF=7$FeimF (7&FiimFdim7$$\"1++D1*3`i\"F\\s$\"1.XIX^%pm#F=7$F\\jmF(7&F`jmF[jm7$$\" 1LLL$*zym;F\\s$\"1Ax'*fzjKDF=7$FcjmF(7&FgjmFbjm7$$\"1LL$3N1#4#HXFHg0#F=7$F_\\nF(7&Fc\\nF^\\n7$$\"1LLL`v&Q(=F \\s$\"1!eL3Krt&>F=7$Ff\\nF(7&Fj\\nFe\\n7$$\"1++DOl5;>F\\s$\"1k\"f0R2w& =F=7$F]]nF(7&Fa]nF\\]n7$$\"1++v.Uac>F\\s$\"1>B/u)=rw\"F=7$Fd]nF(7&Fh]n Fc]n7$$\"\"#F($\"1+#Qx\\b]n\"F=7$F[^nF(7\"-%&STYLEG6#%,PATCHNOGRIDG-F \\bl6&F^bl$\"*++++\"!\")Fg^nF(-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(F gal%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "F ind the value of the yellow shaded area, the area under the F(7,10) de nsity curve to the left of 2." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(F(2));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+$*p'\\X)!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The value of this area ca n also be found by using the integration command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(int(f (x),x=0..2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$*p'\\X)!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a:=plot(f(x),x=0..10):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "b:=plot(f(x),x=1..4,color=ye llow,filled=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "displ ay([a,b]);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 99 "Find the value of the yellow shaded area, the area under the F(7,10) density curve between 1 and 4." }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7do7$\"\"!F(7$$\"1LL$ 3FWYs#!#<$\"1VA&pHT'p_!#=7$$\"1mmmT&)G\\aF,$\"1d%)>a%zia#F,7$$\"1++]7G $R<)F,$\"1avw^Sx5gF,7$$\"1LLL3x&)*3\"!#;$\"1'4'3eZ#*f5F=7$$\"1mmTN@Ki8 F=$\"1U%G:v.[f\"F=7$$\"1++]ilyM;F=$\"1.l$)GbQs@F=7$$\"1LLe*)4D2>F=$\"1 a\"z-k(ykFF=7$$\"1nmm;arz@F=$\"1V0Lk,+]LF=7$$\"1+D\"yD&y;CF=$\"1d!Rj\" \\LSQF=7$$\"1L$e*)4bQl#F=$\"1Ri)zrkYI%F=7$$\"1mT5S\\#4*GF=$\"1)oSq%>LP ZF=7$$\"1++D\"y%*z7$F=$\"1>Zt)[(eM^F=7$$\"1m;ajW8-OF=$\"1;LF!z,`\"eF=7 $$\"1LL$e9ui2%F=$\"1ViqWWXUjF=7$$\"1n;H2Q\\4YF=$\"10*e&QwwhnF=7$$\"1++ voMrU^F=$\"1a*o$Q]TT8(F=7$$\"1+v$4'HaUfF=$\"1lea!*)e#[rF=7$$\"1nmm\"z_\"4i F=$\"1=*[bMqp8(F=7$$\"1nmmm6m#G(F=$\"1._.#y\"f*)oF=7$$\"1ommT&phN)F=$ \"1P&e-w3&HkF=7$$\"1,+v=ddC%*F=$\"1!oykb(epeF=7$$\"1LLe*=)H\\5!#:$\"1a fMOWLy_F=7$$\"1++v=JN[6F\\s$\"1i#[Y)zeRZF=7$$\"1nm\"z/3uC\"F\\s$\"1^q' 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$$\"1++](4&G]F=7$Fgh lF(7&F[ilFfhl7$$\"1+]PpnsM>F\\s$\"1;G^,yM:=F=7$F^ilF(7&FbilF]il7$$\"1+ +]siL-?F\\s$\"1Pa)=k_-n\"F=7$FeilF(7&FiilFdil7$$\"1+++!R5'f?F\\s$\"15a JKH*pb\"F=7$F\\jlF(7&F`jlF[jl7$$\"1+]P/QBE@F\\s$\"1C+K+6_N9F=7$FcjlF(7 &FgjlFbjl7$$\"1+++:o?&=#F\\s$\"1n#>St=lL\"F=7$FjjlF(7&F^[mFijl7$$\"1+] Pa&4*\\AF\\s$\"18m'[)\\OO7F=7$Fa[mF(7&Fe[mF`[m7$$\"1+]7j=_6BF\\s$\"1'[ 9@ly&[6F=7$Fh[mF(7&F\\\\mFg[m7$$\"1++vVy!eP#F\\s$\"1fcr\"GAU1\"F=7$F_ \\mF(7&Fc\\mF^\\m7$$\"1+](=WU[V#F\\s$\"1_$4.reu#**F,7$Ff\\mF(7&Fj\\mFe \\m7$$\"1++DJ#>&)\\#F\\s$\"1<%HPIHb@*F,7$F]]mF(7&Fa]mF\\]m7$$\"1+]P>:m kDF\\s$\"1YM?.6cN&)F,7$Fd]mF(7&Fh]mFc]m7$$\"1+]iv&QAi#F\\s$\"1v([/\"R( *))zF,7$F[^mF(7&F_^mFj]m7$$\"1++vtLU%o#F\\s$\"1[k<\\+dM%Hs%G%F,7$FjamF(7&F^ bmFiam7$$\"1+]P/-a[KF\\s$\"1XBw7z#4-%F,7$FabmF(7&FebmF`bm7$$\"1+](=Yb; J$F\\s$\"1!*))zqZvkPF,7$FhbmF(7&F\\cmFgbm7$$\"1++]i@OtLF\\s$\"1P'[(p\" H@`$F,7$F_cmF(7&FccmF^cm7$$\"1+]PfL'zV$F\\s$\"1jALHc)fI$F,7$FfcmF(7&Fj cmFecm7$$\"1+++!*>=+NF\\s$\"1e&y1b&o.JF,7$F]dmF(7&FadmF\\dm7$$\"1++DE& 4Qc$F\\s$\"1D6f10I6HF,7$FddmF(7&FhdmFcdm7$$\"1+]P%>5pi$F\\s$\"1v@//Q'Q t#F,7$F[emF(7&F_emFjdm7$$\"1+++bJ*[o$F\\s$\"1'f7(4*f;e#F,7$FbemF(7&Ffe mFaem7$$\"1++Dr\"[8v$F\\s$\"1,/Gd&=!>CF,7$FiemF(7&F]fmFhem7$$\"1+++Ijy 5QF\\s$\"1I_+V#RMG#F,7$F`fmF(7&FdfmF_fm7$$\"1+]P/)fT(QF\\s$\"1QYPp>R[@ F,7$FgfmF(7&F[gmFffm7$$\"1+]i0j\"[$RF\\s$\"1c$3%4nlF?F,7$F^gmF(7&FbgmF ]gm7$$\"\"%F($\"131Lv.^1>F,7$FegmF(7\"-%&STYLEG6#%,PATCHNOGRIDG-F\\bl6 &F^bl$\"*++++\"!\")FahmF(-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fgal%( DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Us ing the CDF, we find the area to the left of 4 and then subtract the a rea to the left of 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "area1:=F(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&area1G$\"+@gzf(*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "area2:=F(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ar ea2G$\"+*)[(f;&!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "area_ between:=area1-area2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-area_betwe enG$\"+K6#Qf%!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 74 "Using the integration command, we can also integra te the FPDF from 1 to 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(int(f(x),x=1..4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K6#Qf%!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "a:=plot(f(x),x=0..10,labels=[\" \",\" \"]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "b:=plot(f(x),x=1.5..10,color =yellow,filled=true,labels=[\" \",\" \"]):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 39 "c:=textplot([1.5,-.04,\"x\"],color=blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "d:=textplot([2,0.08,\".25\"] ,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display([ a,b,c,d]);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 18 "Find the value of " }{TEXT 276 1 "x" }{TEXT -1 3 " . 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0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "with(plots, display):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:=x->FPDF(nu[1],nu[2],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%%FPDFG6%&%#nuG6#\"\"\"&F06#\"\"# 9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "EX:=int(x*f(x), x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EXG,$*&&%#nuG6# \"\"#\"\"\",&F'#\"\"\"F*!\"\"F.!\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The expected value given \+ above is only valid for " }{XPPEDIT 18 0 "nu[2]>2" "6#2\"\"#&%#nuG6#\" \"#" }{TEXT -1 46 ", otherwise the expected value does not exist." }} {PARA 0 "" 0 "" {TEXT -1 40 "Calculating Var(X), the variance of the \+ " }{TEXT 264 1 "F" }{TEXT -1 1 "(" }{TEXT 262 0 "" }{XPPEDIT 18 0 "nu[ 1],nu[2];" "6$&%#nuG6#\"\"\"&F$6#\"\"#" }{TEXT -1 57 ") distribution. \+ We will employ the formula: Var(X) = E( " }{TEXT 265 0 "" }{XPPEDIT 267 0 "X^2;" "6#*$)%\"XG\"\"#\"\"\"" }{TEXT -1 5 " ) - " }{XPPEDIT 268 0 "E(X)^2;" "6#*$-%\"EG6#%\"XG\"\"#" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "E_X_SQ:=int((x^2) *f(x),x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'E_X_SQG,$* &*&)&%#nuG6#\"\"#F,\"\"\",&&F*6#\"\"\"#F1F,F1F1F1F-*(F/\"\"\",&F)F2!\" #F1\"\"\",&F)F2!\"\"F1\"\"\"!\"\"F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "VarX:=simplify(E_X_SQ-EX^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VarXG,$*&*&)&%#nuG6#\"\"#F,\"\"\",(&F*6#\"\"\"F1F)F1 !\"#F1F1F-*(F/\"\"\",&F)F1!\"%F1\"\"\"),&F)F1F2F1\"\"#F-!\"\"F," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Th e variance given above is only valid for " }{XPPEDIT 18 0 "4 < nu[2]; " "6#2\"\"%&%#nuG6#\"\"#" }{TEXT -1 40 ", otherwise the variance does \+ not exist." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Simulation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 275 "We will now simulate som e data from an F distribution in order to compare the sampling distrib ution to the theoretical distribution. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "theo_mean:=nu[2]/(nu[2]-2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*theo_meanG#\"\"&\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sim_mean:=Mean(sample);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)sim_meanG$\"+)R0(>8!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "relative_error:=(sim_mean-theo_mean )/theo_mean;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/relative_errorG$\"+ S=Vwb!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "3 " 0 }{VIEWOPTS 1 1 0 1 1 1803 }