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However, you are encouraged to try other values of " }{XPPEDIT 18 0 "nu;" "6#%#nuG" }{TEXT -1 1 "." }}{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 6 "nu:=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nuG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f:=x->ChisquarePDF(nu,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f GR6#%\"xG6\"6$%)operatorG%&arrowGF(-%-ChisquarePDFG6$%#nuG9$F(F(F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=0..20);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7bo7 $\"\"!F(7$$\"1LLL3x&)*3\"!#;$\"13b@_x#3`%!#=7$$\"1nmm;arz@F,$\"1O&o#ep a87!#<7$$\"1+++DJdpKF,$\"1s)4$[4>6@F57$$\"1LLLL3VfVF,$\"1b>Y2u+yIF57$$ \"1mm\"z>5xI&F,$\"11n^]n&F57$$\"1nmm\"H[D:)F,$\"1/2;zBw6lF57$$\"1MLe9 w)*=#*F,$\"1g9P[NzBuF57$$\"1++v$pU&G5!#:$\"1T'>cO\\UH)F57$$\"1n;/Em=N6 Ffn$\"1L;b6rt<\"*F57$$\"1LLLe0$=C\"Ffn$\"15RA`5f!*)*F57$$\"1LLLLA`c9Ff n$\"1Hb8s@YG6F,7$$\"1MLL3RBr;Ffn$\"1*\\Oc^ydC\"F,7$$\"1++vV^\"\\)=Ffn$ \"1(QBtG!)4M\"F,7$$\"1nm;zjf)4#Ffn$\"1fA\"HL@dT\"F,7$$\"1++]Piq'H#Ffn$ \"1K?>v:*zY\"F,7$$\"1LL$e4;[\\#Ffn$\"19X>uXB0:F,7$$\"1++]P:F,7$$\"1nm;zt%**p#Ffn$\"1!)=$RC^%H:F,7$$\"1ML$3-8D!GFfn$\"1>!) 4A)pl`\"F,7$$\"1++]i'y]!HFfn$\"1eq(H4@1a\"F,7$$\"1nm;Ha6eHFfn$\"1UPj$z v:a\"F,7$$\"1ML$e>_6,$Ffn$\"1wk\"zN(yT:F,7$$\"1++]i*)=kIFfn$\"1kuLP9GT :F,7$$\"1nm;HdA[;+5d`\"F,7$ $\"1ML$ezs$HLFfn$\"1*y`(o@')G:F,7$$\"1nm;/&R3a$Ffn$\"1%zO\\9n&3:F,7$$ \"1++]7iI_PFfn$\"1!)Q^sZe![\"F,7$$\"1nmm;_M(=%Ffn$\"1HD6??>/9F,7$$\"1M LL3y_qXFfn$\"1+Q+b%*3A8F,7$$\"1,++]1!>+&Ffn$\"1GU/`7&*>7F,7$$\"1+++]Z/ NaFfn$\"1(yB@uEF6\"F,7$$\"1+++]$fC&eFfn$\"1!3'fm!R\"45F,7$$\"1ML$ez6:B 'Ffn$\"1C$[2a_K<*F57$$\"1nmm;=C#o'Ffn$\"1!p(yA!e48)F57$$\"1nmmm#pS1(Ff n$\"1-vE,*ot2kF57$$\"1mmmm(y8!zFfn$\"1 s?`bw.$o&F57$$\"1,+]i.tK$)Ffn$\"1,$)yv1ng\\F57$$\"1,+](3zMu)Ffn$\"1:%) >Y/.UVF57$$\"1omm\"H_?<*Ffn$\"1,;'[SF`w$F57$$\"1nm;zihl&*Ffn$\"1tSXO\\ !RH$F57$$\"1LLL3#G,***Ffn$\"1^v*eLgK%GF57$$\"1LLezw5V5!#9$\"1Nj/H5MLCF 57$$\"1++v$Q#\\\"3\"F[z$\"1'[4*3'z-7#F57$$\"1LL$e\"*[H7\"F[z$\"1`xmafP B=F57$$\"1+++qvxl6F[z$\"1$o:a8#*ob\"F57$$\"1++]_qn27F[z$\"1yD;:_IJ8F57 $$\"1++Dcp@[7F[z$\"1*f7)GWBU6F57$$\"1++]2'HKH\"F[z$\"1T_+6E6='*F/7$$\" 1nmmwanL8F[z$\"1:7b5OdG#)F/7$$\"1+++v+'oP\"F[z$\"1@q?XP?bpF/7$$\"1LLeR <*fT\"F[z$\"1+=#G$*eZ'fF/7$$\"1+++&)Hxe9F[z$\"1(>4Lo#*f.&F/7$$\"1nm\"H !o-*\\\"F[z$\"1`P*e\")3&*G%F/7$$\"1++DTO5T:F[z$\"1$*=/>^.BOF/7$$\"1nmm T9C#e\"F[z$\"1_vCReMoIF/7$$\"1++D1*3`i\"F[z$\"1Q4`m[gvDF/7$$\"1MLL$*zy m;F[z$\"1n1&))=GQ<#F/7$$\"1ML$3N1#47$$\"1++DOl5;> F[z$\"1^-2u,v-xFb`l7$$\"1++v.Uac>F[z$\"1rL!HTUH\\'Fb`l7$$\"#?F($\"1UFR P1%**R&Fb`l-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"%!G- %%VIEWG6$;F(F^al%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 21 "To see the effect of " }{XPPEDIT 18 0 "nu;" "6#%#nuG" } {TEXT -1 21 " on the shape of the " }{XPPEDIT 18 0 "chi^2;" "6#*$%$chi G\"\"#" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "nu;" "6#%#nuG" }{TEXT -1 70 ") distribution, the following animation will draw a series of PDFs as \+ " }{XPPEDIT 18 0 "nu;" "6#%#nuG" }{TEXT -1 58 " varies from 1 to 21 by increments of 1. What happens as " }{XPPEDIT 18 0 "nu;" "6#%#nuG" } {TEXT -1 11 " increases?" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "nu[0]:=1; nu_step:=1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(nu_stepG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for n from 0 to 20 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "density[n]:=plot(ChisquarePDF(nu[0]+n*nu_step,x),x=0..35):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "num:=convert(evalf(nu[0]+n*nu_step, 3),string):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "tracker[n]:=textplot ([18,0.3,`nu is `.num],color=blue);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "P[n]:=display(\{density[n],tracker[n]\}):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display([seq (P[n], n=0..20)], insequence=true);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6#-%(ANIMAT 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0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 88 "The cumulative distribution function (CDF ) may be used to compute probabilities for the " }{XPPEDIT 18 0 "chi^2 ;" "6#*$%$chiG\"\"#" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "nu;" "6#%#nuG" } {TEXT -1 29 ") distribution. The CDF for " }{XPPEDIT 18 0 "chi^2;" "6 #*$%$chiG\"\"#" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "5;" "6#\"\"&" }{TEXT -1 64 ") is shown below, but you are encouraged to try other values of " }{XPPEDIT 18 0 "nu" "6#%#nuG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: wi th(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "nu:=5;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nuG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(ChisquareCDF(nu,x),x=0..20);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7U7$\"\"!F(7$$\" 1nmm;arz@!#;$\"1'z)>8!#<7$$\"1nmm\"H[D:)F,$\"1tB,mW)\\R#F:7$$\"1L LLe0$=C\"!#:$\"1K[fF:7$$\"1MLL3RBr;FC$\"1\\[0#>j\\2\"F,7$$\"1nm ;zjf)4#FC$\"1(Q/#f&*eY;F,7$$\"1LL$e4;[\\#FC$\"1N%3$eUCFAF,7$$\"1++]i'y ]!HFC$\"1ovc@)GQ&GF,7$$\"1ML$ezs$HLFC$\"1:@c$*=_1NF,7$$\"1++]7iI_PFC$ \"1;_EaP+WTF,7$$\"1nmm;_M(=%FC$\"14Je#p'HsZF,7$$\"1MLL3y_qXFC$\"1&f;MF r\\H&F,7$$\"1,++]1!>+&FC$\"1L-(=F&G)*F,7$$\"1LLeR <*fT\"F\\t$\"1*QnT<^P&)*F,7$$\"1+++&)Hxe9F\\t$\"1Z(z*[!Hs()*F,7$$\"1nm \"H!o-*\\\"F\\t$\"1Hy`x*ff*)*F,7$$\"1++DTO5T:F\\t$\"1#e;p')oD\"**F,7$$ \"1nmmT9C#e\"F\\t$\"19u>3>IE**F,7$$\"1++D1*3`i\"F\\t$\"1\"Q]=fD%Q**F,7 $$\"1MLL$*zym;F\\t$\"1\"\\,u$HD[**F,7$$\"1ML$3N1#4F\\t$\"1h10fB1#)**F,7$$\"1++v.Uac>F\\t$\"1>]Dcd#\\)**F ,7$$\"#?F($\"1&[wLp-v)**F,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABEL SG6$Q\"x6\"%!G-%%VIEWG6$;F(F^[l%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "a:=plot(ChisquarePDF(nu,x),x=0..20,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "b:=plot(ChisquarePDF(nu,x),x=0..6,c olor=yellow,filled=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display([a,b]);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 82 "Find the value of the yellow shaded area, the area under the PDF to the left of 6." }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 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33 "fsolve(ChisquareCDF(nu,x)=.75,x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+k(zci'!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Properties of the Di stribution" }}{EXCHG {PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 282 61 "Calculation of Mean, Variance, and Mom ent Generating Function" }}{PARA 0 "" 0 "" {TEXT -1 29 "To calculate t he mean of the " }{XPPEDIT 18 0 "chi^2;" "6#*$%$chiG\"\"#" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "nu;" "6#%#nuG" }{TEXT -1 46 ") distribution, we can simply integrate over " }{TEXT 283 1 "x" }{TEXT -1 65 "*(Chisquar ePDF), per the definition of mathematical expectation." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "with(plots, displ ay):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f:=x->ChisquarePDF( nu,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operato rG%&arrowGF(-%-ChisquarePDFG6$%#nuG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "EX:=int(x*f(x),x=0..infinity);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#EXG%#nuG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Calculating Var(" }{TEXT 256 1 "X" } {TEXT -1 36 "), we will employ the formula: Var(" }{TEXT 257 1 "X" } {TEXT -1 7 ") = E( " }{XPPEDIT 18 0 "X^2;" "6#*$%\"XG\"\"#" }{TEXT -1 5 " ) - " }{XPPEDIT 18 0 "( E(X) )^2" "6#*$-%\"EG6#%\"XG\"\"#" }} {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "E_X_SQ:=int((x^2)*f(x),x=0..infinity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'E_X_SQG,$*&%#nuG\"\"\",&F'#F(\"\"#F(F(F(F+" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "VarX:=simplify(E_X_SQ-EX^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VarXG,$%#nuG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The Mome nt Generating Function (MGF) can be easily calculated via Maple. Reca ll the moment generating function of a random variable " }{TEXT 275 1 "X" }{TEXT -1 15 " is defined as " }{TEXT 258 1 "M" }{TEXT -1 1 "(" } {TEXT 259 3 " t " }{TEXT -1 6 ") = E(" }{XPPEDIT 18 0 "exp(tX)" "6#-%$ expG6#%#tXG" }{TEXT -1 63 "), provided this expectation exists. In ta king the integral of" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(tx)" "6#-%$expG6#%#txG" }{TEXT -1 35 "*ChisquarePDF(x) over the r ange (0," }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 43 "), t he argument of the exponential term is " }{XPPEDIT 18 0 "x(t-1/2);" "6 #-%\"xG6#,&%\"tG\"\"\"*&\"\"\"F(\"\"#!\"\"F," }{TEXT -1 64 ", and for \+ the resulting integral to be convergent, we must have " }{XPPEDIT 18 0 "x(t-1/2);" "6#-%\"xG6#,&%\"tG\"\"\"*&\"\"\"F(\"\"#!\"\"F," }{TEXT -1 30 "<0. Equivalently, we require " }{XPPEDIT 18 0 "t < 1/2;" "6#2% \"tG*&\"\"\"\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "assume(t<1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "assum e(nu>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "int(exp(t*x)*f( x),x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&)\" \"#,$%$nu|irG#\"\"\"F'\"\"\"),&%#t|irG!\"\"F*F+F(\"\"\"!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "So we have the moment generating function for a " }{XPPEDIT 18 0 "chi^2; " "6#*$%$chiG\"\"#" }{TEXT -1 2 "( " }{XPPEDIT 18 0 "nu;" "6#%#nuG" } {TEXT -1 29 ") random variable is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 260 1 "M" }{TEXT -1 2 "( " }{TEXT 261 1 "t" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "(1-2*t) ^(-nu/2);" "6#),&\"\"\"\"\"\"*&\"\"#F&%\"tGF&!\"\",$*&%#nuGF&\"\"#F*F* " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "We will now define " }{TEXT 262 1 "M" }{TEXT -1 2 "( " } {TEXT 263 1 "t" }{TEXT -1 20 " ) as a function of " }{TEXT 264 1 "t" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "M:=t->(1-2*t)^(-nu/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGR6#%\"tG6\"6$%)operatorG%&arrowGF(),&\"\"\"F.9$!\"#,$%#nuG #!\"\"\"\"#F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 117 "The moment generating function provides us altern ative ways to calculate the mean and variance by way of the formula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " " } {XPPEDIT 18 0 "M^r;" "6#)%\"MG%\"rG" }{TEXT -1 9 " (0) = E(" } {XPPEDIT 18 0 "X^r" "6#)%\"XG%\"rG" }{TEXT -1 10 ") , where " } {XPPEDIT 18 0 "M^r" "6#)%\"MG%\"rG" }{TEXT -1 1 "(" }{TEXT 265 1 "t" } {TEXT -1 14 ") denotes the " }{TEXT 266 1 "r" }{TEXT -1 18 " th deriva tive of " }{TEXT 267 1 "M" }{TEXT -1 1 "(" }{TEXT 268 1 "t" }{TEXT -1 18 ") with respect to " }{TEXT 269 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "This formula holds a s long as " }{TEXT 270 1 "M" }{TEXT -1 1 "(" }{TEXT 271 1 "t" }{TEXT -1 227 ") exists in an open interval containing zero. See, for exampl e, Mathematical Statistics and Data Analysis by John A. Rice for more \+ on the moment generating function. We now use this formula to find th e mean of the Chi-square." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M_p:=diff(M(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M_pG*&*&),&\"\"\"F)%\"tG!\"#,$%#nuG#!\"\" \"\"#F)F-F)\"\"\"F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(M_p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&),&\"\"\"F(% \"tG!\"#,$%#nuG#!\"\"\"\"#F(F,F(\"\"\",&F.F(F)F/!\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify(subs(t=0,M_p));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#nuG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "And therefore, if " }{TEXT 272 1 "X " }{TEXT -1 6 " is a " }{XPPEDIT 18 0 "chi^2;" "6#*$%$chiG\"\"#" } {TEXT -1 1 "(" }{XPPEDIT 18 0 "nu;" "6#%#nuG" }{TEXT -1 19 ") variable , then E(" }{TEXT 273 1 "X" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "nu;" "6 #%#nuG" }{TEXT -1 77 ", which agrees with what we found earlier. Now \+ turning to the second moment." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M_pp:=diff(M_p,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%M_ppG,&*&*&),&\"\"\"F*%\"tG!\"#,$%# nuG#!\"\"\"\"#F*)F.F1\"\"\"F3*$)F)\"\"#F3!\"\"F**&*&F(F3F.F*F3*$)F)\" \"#F3F7F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(subs( t=0,M_pp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%#nuG\"\"#\"\"\"\" \"\"F&F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Therefore, E(" }{XPPEDIT 18 0 "X^2" "6#*$%\"XG\"\"#" } {TEXT -1 94 ") again agrees with what was calculated previously. The \+ variance is now quickly calculated as" }}{PARA 0 "" 0 "" {TEXT -1 4 "V ar(" }{TEXT 274 1 "X" }{TEXT -1 7 ") = E( " }{XPPEDIT 18 0 "X^2;" "6#* $%\"XG\"\"#" }{TEXT -1 5 " ) - " }{XPPEDIT 18 0 "( E(X) )^2" "6#*$-%\" EG6#%\"XG\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 14 " = " } {XPPEDIT 18 0 "nu^2+2*nu-nu^2;" "6#,(*$%#nuG\"\"#\"\"\"*&\"\"#F'F%F'F' *$F%\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 14 " = " } {XPPEDIT 18 0 "2*nu;" "6#*&\"\"#\"\"\"%#nuGF%" }{TEXT -1 3 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 0 "" }} {PARA 264 "" 0 "" {TEXT 284 18 "Special Properties" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "You probably noticed in \+ " }{TEXT 297 43 "Probability Distribution Function and Shape" }{TEXT -1 45 " that the probability density function for a " }{XPPEDIT 18 0 " chi^2;" "6#*$%$chiG\"\"#" }{TEXT -1 2 "( " }{XPPEDIT 18 0 "nu;" "6#%#n uG" }{TEXT -1 64 " ) distribution looked close to Normal (i.e. bell-s haped) when " }{XPPEDIT 18 0 "nu;" "6#%#nuG" }{TEXT -1 50 " was large. Let's look again at the shape of the " }{XPPEDIT 18 0 "chi^2;" "6#*$ %$chiG\"\"#" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "nu;" "6#%#nuG" }{TEXT -1 19 ") distribution, as " }{XPPEDIT 18 0 "nu;" "6#%#nuG" }{TEXT -1 77 " varies from 1 to 25. A normal curve with the same mean and variance a s each " }{XPPEDIT 18 0 "chi^2;" "6#*$%$chiG\"\"#" }{TEXT -1 35 " will be overlayed for comparison. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for nu from 1 to 25 do" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "H[nu]:=plot(ChisquarePDF(nu,x),x=1. .50,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "N[nu]:=plot(Nor malPDF(nu,2*nu,x),x = 1..50):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "nu m:=convert(nu,string):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "tracker[n u]:=textplot([30,0.2,`nu is `.num],color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "P[nu]:=display(\{H[nu],N[nu],tracker[nu]\}):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "display([seq(P[nu], nu=1..25)], insequence=true,title=\"Chisquar e (blue) to Normal (red)\"); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%(ANIMATEG6;7'-%'CURVESG6$7]o7$$\"\"\"\"\"!$\"1\"y Qx\"z%4#G!#;7$$\"1n;aQ^,n7!#:$\"1,I$4/76x#F17$$\"1LL3x-.M:F5$\"1#*H*3) \\#oi#F17$$\"1nTNYy`n;F5$\"1Q\"yophN_#F17$$\"1+]i:a/,=F5$\"1$=H/fZGS#F 17$$\"1Le*[)HbM>F5$\"1>T)oE3wE#F17$$\"1mm;a01o?F5$\"1Hu)fa$*47#F17$$\" 1$e*[)**)Q+BF5$\"1__S8%*R[=F17$$\"1+D\"GWleKF5$\"1Yvh;&H'zy!#<7$$\"1+vo*fH*>NF5$\"1o)R#H(ppw&Fdo7$$\"1$3-QB27y $F5$\"1*)otFL6zSFdo7$$\"1nm\"z'[[USF5$\"1@l-[AV)y#Fdo7$$\"1nmmrR]oXF5$ \"16uL(\\Y*o6Fdo7$$\"1mmTvI_%4&F5$\"1Y!zEPBsE%!#=7$$\"1$3-))e#Gc`F5$\" 1>v_?0baCF^q7$$\"1*\\(=-@/=cF5$\"1Rd,8?Lk8F^q7$$\"1;Hd:;!)zeF5$\"1#=Yo &32Gt!#>7$$\"1K$e*G6cThF5$\"1mPQzlZ.QF^r7$$\"1+](=GIpi'F5$\"1o?-**QnH5 F^r7$$\"1m;zM%*H7rF5$\"1/NfP\"yxZ#!#?7$$\"1**\\7BFW<\")F5$\"1JNetc7=*) !#A7$$\"1n;z\\L'p:*F5$\"1&Q>?%H(Ro\"!#B7$$\"1+D1A]J>5!#9$\"12N#\\:h5)= !#D7$$\"1LL3y&**e7\"F^t$\"1Kg9A7F^t$\"1:_PE1Tfo!# I7$$\"1++DfcYD8F^t$\"1U8P$fPpR\"!#K7$$\"1++vjfeJ9F^t$\"1q+u(Qi6e\"!#N7 $$\"1++vSD&Q`\"F^t$\"1tBaEd&RM\"!#Q7$$\"1n\"z**Q?ni\"F^t$\"1m(yI&*>2R \"!#T7$$\"1LL3X#\\rt\"F^t$\"1q'p_dP!QA!#X7$$\"1LLLqppI=F^t$\"1Rd6tn(e \\)!#\\7$$\"1+D\"Q@:&R>F^t$\"1\"R\"\\!*3NT^!#`7$$\"1LL$yzPe.#F^t$\"1>+ lo=L\"z&!#d7$$\"1+D\")Q*=:9#F^t$\"1wz\"pNF^t$\"1Y9!GJtD-'!$Z\"7$$\"1 ++D8Q*Rn$F^t$\"1&**4Z^jf\"e!$b\"7$$\"1Le9nchsPF^t$\"1aj')y5#>,\"!$i\"7 $$\"1+D1@RqvQF^t$\"1Mocj#R6m%!$r\"7$$\"1LL3K:\\wRF^t$\"1>!o8Yn9(>!$z\" 7$$\"1+DJ?o+#3%F^t$\"1kH@S8ke>!$)=7$$\"1nmm$eIO=%F^t$\"1zBr\\lZpC!$(>7 $$\"1m;afbb(G%F^t$\"1\\/jnHjZ6!$1#7$$\"1Le9%)*>1R%F^t$\"1o!ys!p;QP!$;# 7$$\"1++]'[D`[%F^t$\"1&\\O>WLL[%!$D#7$$\"1L$3(z'oQf%F^t$\"1XA#H1^+a\"! 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do7$Fi]l$\"1kUsm(\\=U\"Fdo7$F_^l$\"1buF%oDq6\"Fdo7$Fe^l$\"1Dv0W9^,&)F^ q7$F[_l$\"1(*zEc;^xjF^q7$Fa_l$\"1(yMtO_%=YF^q7$Fg_l$\"1!*[g&)Q79LF^q7$ F]`l$\"15E63'o/J#F^q7$Fc`l$\"1b\\6=.[\"e\"F^q7$Fi`l$\"1%\\V'RWe&4\"F^q 7$F_al$\"1[iXkNXOqF^r7$Feal$\"1Gd_97!>k%F^r7$F[bl$\"1%ojo]Ey\"HF^r7$Fa bl$\"1a3$>+(zL=F^r7$Fgbl$\"1bj<:@9*3\"F^rF\\cl-Fe`m6%Fg`m%)nu~is~25GF_ `mF]amFcam-%&TITLEG6#QAChisquare~(blue)~to~Normal~(red)Faam" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Indeed, the " }{XPPEDIT 18 0 "chi^2;" "6#*$%$chiG\"\"#" }{TEXT -1 1 "(" } {XPPEDIT 18 0 "nu;" "6#%#nuG" }{TEXT -1 35 ") distribution approaches \+ a Normal(" }{XPPEDIT 18 0 "nu,2*nu;" "6$%#nuG*&\"\"#\"\"\"F#F&" } {TEXT -1 18 ") distribution as " }{XPPEDIT 18 0 "nu;" "6#%#nuG" } {TEXT -1 57 " grows large. This can also be seen by realizing that if \+ " }{TEXT 285 1 "Y" }{TEXT -1 7 " is a " }{XPPEDIT 18 0 "chi^2;" "6#*$ %$chiG\"\"#" }{TEXT -1 15 "variable, then " }{TEXT 286 1 "Y" }{TEXT -1 30 " has the same distribution as " }{XPPEDIT 287 0 "X[1]+X[2];" "6 #,&&%\"XG6#\"\"\"\"\"\"&F%6#\"\"#F(" }{TEXT -1 7 "+ ... +" }{XPPEDIT 18 0 "X[nu];" "6#&%\"XG6#%#nuG" }{TEXT -1 12 " where the " }{XPPEDIT 288 0 "X[i];" "6#&%\"XG6#%\"iG" }{TEXT -1 48 " 's are independent and \+ identically distributed " }{XPPEDIT 18 0 "chi^2;" "6#*$%$chiG\"\"#" } {TEXT -1 8 "(1) for " }{XPPEDIT 18 0 "i <= nu;" "6#1%\"iG%#nuG" } {TEXT -1 66 ". By the Central Limit Theorem, we know that the distrib ution of " }{XPPEDIT 289 0 "X[1]+X[2];" "6#,&&%\"XG6#\"\"\"\"\"\"&F%6# \"\"#F(" }{TEXT -1 8 "+ ... + " }{XPPEDIT 18 0 "X[nu];" "6#&%\"XG6#%#n uG" }{TEXT -1 25 " approaches normality as " }{XPPEDIT 18 0 "nu" "6#%# nuG" }{TEXT -1 63 " becomes larger, and therefore, so does the distrib ution of a " }{XPPEDIT 18 0 "chi^2;" "6#*$%$chiG\"\"#" }{TEXT -1 1 "( " }{XPPEDIT 18 0 "nu;" "6#%#nuG" }{TEXT -1 14 " ) variable. \n" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Simulation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 212 "We will now simulat e some data from a Chi-square distribution in order to compare the sam pling distribution to the theoretical distribution. The next bit of c ode will simulate a random sample of size 500 from a " }{XPPEDIT 18 0 "chi^2;" "6#*$%$chiG\"\"#" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "5;" "6#\"\" &" }{TEXT -1 62 ") distribution, but you are encouraged to change the \+ value of " }{XPPEDIT 18 0 "nu" "6#%#nuG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restar t: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "nu:=5: n :=500:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sample:=Chisquare S(nu, n):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "Remember that when you plot histograms, the number of bi ns that you select can make a huge difference in the plot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "num _bins:=20:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a:=Histogram( sample,0..20,num_bins):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " b:=plot(ChisquarePDF(nu,x),x=0..20,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display([a,b],title=\"Simulated vs. Theoretic al\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CUR VESG6%7\\p7$\"\"!F(7$F($\"1+++++++Y!#<7$$\"\"\"F(F*7$F.F(F07$F.$\"1+++ +++!G\"!#;7$$\"\"#F(F27$F6F(F87$F6$\"1++++++g7F47$$\"\"$F(F:7$F=F(F?7$ F=$\"1++++++?;F47$$\"\"%F(FA7$FDF(FF7$FD$\"1++++++?8F47$$\"\"&F(FH7$FK F(FM7$FK$\"1+++++++#*F,7$$\"\"'F(FO7$FRF(FT7$FR$\"1*************f)F,7$ $\"\"(F(FV7$FYF(Fen7$FY$\"1+++++++aF,7$$\"\")F(Fgn7$FjnF(F\\o7$Fjn$\"1 +++++++]F,7$$\"\"*F(F^o7$FaoF(Fco7$Fao$\"1+++++++MF,7$$\"#5F(Feo7$FhoF (Fjo7$Fho$\"1+++++++GF,7$$\"#6F(F\\p7$F_pF(Fap7$F_p$\"1+++++++EF,7$$\" #7F(Fcp7$FfpF(Fhp7$Ffp$\"1+++++++AF,7$$\"#8F(Fjp7$F]qF(F_q7$F]q$\"1+++ ++++g!#=7$$\"#9F(Faq7$FeqF(Fgq7$Feq$\"1+++++++SFcq7$$\"#:F(Fiq7$F\\rF( F^r7$F\\r$\"1+++++++?Fcq7$$\"#;F(F`r7$FcrF(FerFbr7$$\"#F(F(F]sF]sF]s7$$\"#?F(F(F`s-%'COLOURG6& %$RGBG$\"*++++\"!\")F(F(-%*LINESTYLEG6#F>-F$6$7boF'7$$\"1LLL3x&)*3\"F4 $\"13b@_x#3`%Fcq7$$\"1nmm;arz@F4$\"1O&o#epa87F,7$$\"1+++DJdpKF4$\"1s)4 $[4>6@F,7$$\"1LLLL3VfVF4$\"1b>Y2u+yIF,7$$\"1mm\"z>5xI&F4$\"11n^]n&F,7 $$\"1nmm\"H[D:)F4$\"1/2;zBw6lF,7$$\"1MLe9w)*=#*F4$\"1g9P[NzBuF,7$$\"1+ +v$pU&G5!#:$\"1T'>cO\\UH)F,7$$\"1n;/Em=N6F`w$\"1L;b6rt<\"*F,7$$\"1LLLe 0$=C\"F`w$\"15RA`5f!*)*F,7$$\"1LLLLA`c9F`w$\"1Hb8s@YG6F47$$\"1MLL3RBr; F`w$\"1*\\Oc^ydC\"F47$$\"1++vV^\"\\)=F`w$\"1(QBtG!)4M\"F47$$\"1nm;zjf) 4#F`w$\"1fA\"HL@dT\"F47$$\"1++]Piq'H#F`w$\"1K?>v:*zY\"F47$$\"1LL$e4;[ \\#F`w$\"19X>uXB0:F47$$\"1++]P:F47$$\"1nm;zt%**p#F` w$\"1!)=$RC^%H:F47$$\"1ML$3-8D!GF`w$\"1>!)4A)pl`\"F47$$\"1++]i'y]!HF`w $\"1eq(H4@1a\"F47$$\"1nm;Ha6eHF`w$\"1UPj$zv:a\"F47$$\"1ML$e>_6,$F`w$\" 1wk\"zN(yT:F47$$\"1++]i*)=kIF`w$\"1kuLP9GT:F47$$\"1nm;HdA[;+5d`\"F47$$\"1ML$ezs$HLF`w$\"1*y`(o@') G:F47$$\"1nm;/&R3a$F`w$\"1%zO\\9n&3:F47$$\"1++]7iI_PF`w$\"1!)Q^sZe![\" F47$$\"1nmm;_M(=%F`w$\"1HD6??>/9F47$$\"1MLL3y_qXF`w$\"1+Q+b%*3A8F47$$ \"1,++]1!>+&F`w$\"1GU/`7&*>7F47$$\"1+++]Z/NaF`w$\"1(yB@uEF6\"F47$$\"1+ ++]$fC&eF`w$\"1!3'fm!R\"45F47$$\"1ML$ez6:B'F`w$\"1C$[2a_K<*F,7$$\"1nmm ;=C#o'F`w$\"1!p(yA!e48)F,7$$\"1nmmm#pS1(F`w$\"1-vE,*ot2kF,7$$\"1mmmm(y8!zF`w$\"1s?`bw.$o&F,7$$\"1,+]i.tK$)F` w$\"1,$)yv1ng\\F,7$$\"1,+](3zMu)F`w$\"1:%)>Y/.UVF,7$$\"1omm\"H_?<*F`w$ \"1,;'[SF`w$F,7$$\"1nm;zihl&*F`w$\"1tSXO\\!RH$F,7$$\"1LLL3#G,***F`w$\" 1^v*eLgK%GF,7$$\"1LLezw5V5!#9$\"1Nj/H5MLCF,7$$\"1++v$Q#\\\"3\"Febl$\"1 '[4*3'z-7#F,7$$\"1LL$e\"*[H7\"Febl$\"1`xmafPB=F,7$$\"1+++qvxl6Febl$\"1 $o:a8#*ob\"F,7$$\"1++]_qn27Febl$\"1yD;:_IJ8F,7$$\"1++Dcp@[7Febl$\"1*f7 )GWBU6F,7$$\"1++]2'HKH\"Febl$\"1T_+6E6='*Fcq7$$\"1nmmwanL8Febl$\"1:7b5 OdG#)Fcq7$$\"1+++v+'oP\"Febl$\"1@q?XP?bpFcq7$$\"1LLeR<*fT\"Febl$\"1+=# G$*eZ'fFcq7$$\"1+++&)Hxe9Febl$\"1(>4Lo#*f.&Fcq7$$\"1nm\"H!o-*\\\"Febl$ \"1`P*e\")3&*G%Fcq7$$\"1++DTO5T:Febl$\"1$*=/>^.BOFcq7$$\"1nmmT9C#e\"Fe bl$\"1_vCReMoIFcq7$$\"1++D1*3`i\"Febl$\"1Q4`m[gvDFcq7$$\"1MLL$*zym;Feb l$\"1n1&))=GQ<#Fcq7$$\"1ML$3N1#47$$\"1++D Ol5;>Febl$\"1^-2u,v-xF\\il7$$\"1++v.Uac>Febl$\"1rL!HTUH\\'F\\il7$Fas$ \"1UFRP1%**R&F\\il-Fds6&FfsF(F(F(-%+AXESLABELSG6$%!GF_jl-%&TITLEG6#Q:S imulated~vs.~Theoretical6\"-%%VIEWG6$;F(Fas%(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 63 "Now let's compare the simulated mean to the theoretical mean. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "theo_mean:=nu;" }}{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$\"+91r\")\\!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "relative_error:=(sim_mean-theo_mean)/theo_mean;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%/relative_errorG$!++s(yl$!#7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "3" 0 } {VIEWOPTS 1 1 0 1 1 1803 }