{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 "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 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 "" 0 1 0 0 0 0 1 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 "" -1 263 "" 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 "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 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 "" -1 275 "" 1 18 0 0 0 0 0 1 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 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 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 Ou tput" 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 "" 0 257 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 "" 0 259 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 258 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 275 22 "The Gamma Distribution" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "A random variable" } {TEXT 276 2 " X" }{TEXT -1 13 " has a Gamma(" }{XPPEDIT 18 0 "alpha" " 6#%&alphaG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 62 ") distribution if and only its probability density is given by " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 " \+ " }{XPPEDIT 18 0 "x^(alpha-1)*exp(-x/beta)/(beta^alpha*Ga mma(alpha));" "6#*()%\"xG,&%&alphaG\"\"\"\"\"\"!\"\"F(-%$expG6#,$*&F%F (%%betaGF*F*F(*&)F0F'F(-%&GammaG6#F'F(F*" }{TEXT -1 5 " for " }{TEXT 278 1 "x" }{TEXT -1 0 "" }{TEXT 277 1 " " }{TEXT -1 4 "> 0 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 279 1 "f" }{TEXT -1 2 "( " }{TEXT 280 1 "x" }{TEXT -1 5 " ) = " }}{PARA 0 "" 0 "" {TEXT -1 60 " \+ 0 elsewhere" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "al pha,beta;" "6$%&alphaG%%betaG" }{TEXT -1 5 " > 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The following code will d raw the density function for the Gamma(" }{XPPEDIT 18 0 "alpha,beta;" "6$%&alphaG%%betaG" }{TEXT -1 19 ") distribution for " }{XPPEDIT 18 0 "alpha = 2,beta = 3;" "6$/%&alphaG\"\"#/%%betaG\"\"$" }{TEXT -1 105 ", but you are encouraged to try other values. Remember to make sure th at they are both positive numbers." }}{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 18 "alpha:=2; beta:=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%be taG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f(x):=GammaPDF( alpha,beta,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#%\"xG,$*&F' \"\"\"-%$expG6#,$F'#!\"\"\"\"$F*#F*\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot(f(x),x=0..20, title=\"Gamma(2,3) PDF\");" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7^o7 $\"\"!F(7$$\"1LLL3x&)*3\"!#;$\"1UX!=H]x;\"!#<7$$\"1nmm;arz@F,$\"1[Q2.y <_AF/7$$\"1+++DJdpKF,$\"1R!4taTxD$F/7$$\"1LLLL3VfVF,$\"15g$zU)o)=%F/7$ $\"1mm\"z>5xI&F,$\"1@E)fNY6%\\F/7$$\"1****\\i&*)fD'F,$\"1O\\.ZOsUcF/7$ $\"1LL3F*oU?(F,$\"1zR!R6eeH'F/7$$\"1nmm\"H[D:)F,$\"1)[@T7&)G!pF/7$$\"1 ++v$pU&G5!#:$\"12C*Gvg66)F/7$$\"1LLLe0$=C\"FV$\"1yX5?Q3@\"*F/7$$\"1LLL LA`c9FV$\"1?[v!fl\"f**F/7$$\"1MLL3RBr;FV$\"1hXc&>(zj5F,7$$\"1++vV^\"\\ )=FV$\"1P&oL$fK<6F,7$$\"1nm;zjf)4#FV$\"1ktUnkYe6F,7$$\"1++]Piq'H#FV$\" 1ztqG&3o=\"F,7$$\"1LL$e4;[\\#FV$\"1`hO8vz17F,7$$\"1++]P=97F,7$$\"1nm;zt%**p#FV$\"1[v(=g1(>7F,7$$\"1ML$3-8D!GFV$\"19rQ(G)[B7 F,7$$\"1++]i'y]!HFV$\"1Y(fN)yjD7F,7$$\"1ML$e>_6,$FV$\"1t['>NciA\"F,7$$ \"1nm;HdA7F,7$$\"1nm;/&R3a$FV$\"1X*H*z!y&37F,7$$\"1++]7iI_P FV$\"1bFDOie$>\"F,7$$\"1nmm;_M(=%FV$\"12^au$o@:\"F,7$$\"1MLL3y_qXFV$\" 1#)R29x!o5\"F,7$$\"1,++]1!>+&FV$\"1C$3n%H/\\5F,7$$\"1+++]Z/NaFV$\"1,R% *yxOm)*F/7$$\"1+++]$fC&eFV$\"1owmm67W#*F/7$$\"1ML$ez6:B'FV$\"1!ejDcaXn )F/7$$\"1nmm;=C#o'FV$\"1&\\-vY]V+)F/7$$\"1nmmm#pS1(FV$\"1p/%>.q/X(F/7$ $\"1++]i`A3vFV$\"1_rt<(o\"HoF/7$$\"1mmmm(y8!zFV$\"1e^XP`./jF/7$$\"1,+] i.tK$)FV$\"1Nn#e9Eyv&F/7$$\"1,+](3zMu)FV$\"1CM1f$z&o_F/7$$\"1omm\"H_?< *FV$\"1_'HYE!3\"z%F/7$$\"1nm;zihl&*FV$\"1aV%f%eL#Q%F/7$$\"1LLL3#G,***F V$\"1d=^ge\"H(RF/7$$\"1LLezw5V5!#9$\"1c=w\\9C\"e$F/7$$\"1++v$Q#\\\"3\" Ffx$\"1S'\\Ib%3nKF/7$$\"1LL$e\"*[H7\"Ffx$\"1$>**>P*[aHF/7$$\"1+++qvxl6 Ffx$\"1BI4,h6fEF/7$$\"1++]_qn27Ffx$\"1>DyVKh&R#F/7$$\"1++Dcp@[7Ffx$\"1 y='zSfI;#F/7$$\"1++]2'HKH\"Ffx$\"1Sgv/#>)G>F/7$$\"1nmmwanL8Ffx$\"1Vn#f nf#Q&o-9F/7$$\"1+++&)Hxe9Ffx$\"15J,l8,`7F/7$$\"1nm\"H!o-*\\\"Ffx$\"1l@M#4 4f7\"F/7$$\"1++DTO5T:Ffx$\"1Ej#[^Ng+\"F/7$$\"1nmmT9C#e\"Ffx$\"1@LQ$eh` +*!#=7$$\"1++D1*3`i\"Ffx$\"17eq!eFfx$ \"1(o)[9nn$e$Fj\\l7$$\"1++v.Uac>Ffx$\"1^R?Y!fy>$Fj\\l7$$\"#?F($\"1l%)3 9^2GGFj\\l-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%&TITLEG6#Q/Gamma(2,3)~PDF6 \"-%+AXESLABELSG6$Q\"xFh`l%!G-%%VIEWG6$;F(Fi_l%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "To see the effect of " } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 27 " on the shape of the Gamma(" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 1 "," } {XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 95 ") distribution, the fo llowing animation will draw a series of probability density functions \+ as " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 59 " varies from 0 .1 to 2.1 by increments of 0.1 while holding " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 32 " constant at 3. Do you see why " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 24 " is a \"shape\" parameter?" } }{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "alpha[0]:=0.1; alpha_step:=0.1; beta:=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6#\"\"!$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+alpha_stepG$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for n from 0 to 20 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "dens ity[n]:=plot(GammaPDF(alpha[0]+n*alpha_step,beta,x),x=0..20):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "num:=convert(evalf(alpha[0]+n*alpha _step,3),string):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "tracker[n]:=te xtplot([14,0.3,`alpha is `.num],color=blue);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "P[n]:=display(\{density[n],tracker[n]\}):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "di splay([seq(P[n], n=0..20)], insequence=true, title=\"Ranging Alpha\"); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%(ANIMATEG6 77&-%'CURVESG6$7hn7$$\"1+++M3VfV!#;$\"17K[>!)H>@5xI&F.$\"1='>W,%Q&R\"F.7$$\"1+++#))\\=y&F.$\"1*3'HNXqr7 F.7$$\"1+++j&*)fD'F.$\"1.?[:q/m6F.7$$\"1,++G*oU?(F.$\"1&*[X1l,]**!#<7$ $\"1+++#H[D:)F.$\"16l.ng2D')FT7$$\"1*****fh()*=#*F.$\"1d(*osf,_uFT7$$ \"1+++%pU&G5!#:$\"1R)*[d(Qq^'FT7$$\"1+++Em=N6F\\o$\"1mHPaE3bdFT7$$\"1+ ++e0$=C\"F\\o$\"1u@qe:#H7&FT7$$\"1+++LA`c9F\\o$\"1*o\\;wo98%FT7$$\"1++ +3RBr;F\\o$\"1'3nS%eU)R$FT7$$\"1+++W^\"\\)=F\\o$\"1nW0<#z*RGFT7$$\"1++ +zjf)4#F\\o$\"1M)[$\\))3,CFT7$$\"1+++'4;[\\#F\\o$\"1spp=&R2!=FT7$$\"1+ ++j'y]!HF\\o$\"1RC,\")zYp8FT7$$\"1+++'zs$HLF\\o$\"1SM`Yje^5FT7$$\"1+++ 8iI_PF\\o$\"1O/22&)=,#)!#=7$$\"1+++<_M(=%F\\o$\"1#yuNz-sU'Far7$$\"1+++ 4y_qXF\\o$\"1N5R@C)yA&Far7$$\"1+++]1!>+&F\\o$\"1y9Su/ruTFar7$$\"1+++]Z /NaF\\o$\"1))z&>[7KN$Far7$$\"1+++]$fC&eF\\o$\"1?!GC'RoHFFar7$$\"1+++'z 6:B'F\\o$\"1rsjgbdtAFar7$$\"1+++<=C#o'F\\o$\"1L[q40BP=Far7$$\"1+++n#pS 1(F\\o$\"1+`jMKvQ:Far7$$\"1+++j`A3vF\\o$\"1$fT!fg8c7Far7$$\"1+++n(y8!z F\\o$\"14bY+2Q_5Far7$$\"1+++j.tK$)F\\o$\"1ZKa!f]')o)!#>7$$\"1,++)3zMu) F\\o$\"1sFr%RjdD(Fiu7$$\"1+++#H_?<*F\\o$\"1^*e&o&=Z-'Fiu7$$\"1+++!G;cc *F\\o$\"1j2*zfDz3&Fiu7$$\"1+++4#G,***F\\o$\"16E]w4JZUFiu7$$\"1+++!o2J/ \"!#9$\"1/J`M*Qp_$Fiu7$$\"1+++%Q#\\\"3\"Faw$\"1h&**p`QS+$Fiu7$$\"1+++; *[H7\"Faw$\"1:cpL*G#HDFiu7$$\"1+++qvxl6Faw$\"1GcI:T5?@Fiu7$$\"1+++`qn2 7Faw$\"1Q)oJsvgy\"Fiu7$$\"1+++cp@[7Faw$\"18Tl(eJY^\"Fiu7$$\"1+++3'HKH \"Faw$\"1S*G`D#pi7Fiu7$$\"1+++xanL8Faw$\"1J9%*[,Ft5Fiu7$$\"1+++v+'oP\" Faw$\"1$))*fx&H5.*!#?7$$\"1+++S<*fT\"Faw$\"1OrC%=+#HxF\\z7$$\"1+++&)Hx e9Faw$\"1V)*3oH\"[_'F\\z7$$\"1+++.o-*\\\"Faw$\"1]zv5`VnbF\\z7$$\"1+++T O5T:Faw$\"1O:aW#)y>ZF\\z7$$\"1+++U9C#e\"Faw$\"1fu*R5!e=SF\\z7$$\"1+++1 *3`i\"Faw$\"1%>K,%f/)R$F\\z7$$\"1+++$*zym;Faw$\"1ZI$4H\"*G*GF\\z7$$\"1 +++^j?4Faw$\"10=!z,I:6\"F\\z7$$ \"1+++/Uac>Faw$\"1$p3%>,wK&*!#@7$$\"#?\"\"!$\"1@!p>N7e3)Fc^l-%'COLOURG 6&%$RGBG$\"#5!\"\"Fg^lFg^l-%%TEXTG6%7$$\"#9Fg^l$\"\"$F`_l%,alpha~is~.1 G-F[_l6&F]_lFg^lFg^l$\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$ ;Fg^lFe^l%(DEFAULTG7&-Fb_l6%Fd_l%,alpha~is~.2GFj_l-F(6$7fn7$F,$\"1$HD! 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" }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 62 " w ill vary from 0.2 to 3 at intervals of 0.2. 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hjl$\"1V(>[B=/x$FdzF][m-F(6%F*%,beta~is~3.2GF1Fb[mFh[m-%&TITLEG6#Q-Ran ging~BetaFf[m" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Sample Probability Calculations " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We can calculate probabilities for the Gamma(" }{XPPEDIT 18 0 "alp ha,beta;" "6$%&alphaG%%betaG" }{TEXT -1 146 ") distribution using eith er the probability distribution function (PDF) or the cumulative distr ibution function (CDF). We will denote the PDF by " }{TEXT 283 1 "f" }{TEXT -1 16 " and the CDF by " }{TEXT 284 1 "F" }{TEXT -1 90 ". The \+ CDF for Gamma(2,3) appears below, but feel free to plot it for differe nt values of " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: \+ with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "alpha:=2; b eta:=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot(GammaCDF(alpha,beta,x),x=0..20, title=\"CDF for \+ Gamma(2,3)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 " 6&-%'CURVESG6$7W7$\"\"!F(7$$\"1LLL3x&)*3\"!#;$\"1cZWM67$$\"1nmm; arz@F,$\"1Q^S4g4:D!#=7$$\"1+++DJdpKF,$\"1#)>3kidCbF57$$\"1LLLL3VfVF,$ \"1`%>5oS*)e*F57$$\"1****\\i&*)fD'F,$\"1%yiOI.W*=!#<7$$\"1nmm\"H[D:)F, $\"14n][C*p3$FE7$$\"1LLLe0$=C\"!#:$\"1XRVO*>H`'FE7$$\"1MLL3RBr;FN$\"1H X#=Y-)z5F,7$$\"1nm;zjf)4#FN$\"1;gNCOUc:F,7$$\"1LL$e4;[\\#FN$\"1#G,V;4h 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aa()o)GYf*F,7$$\"1++DTO5T:Fgt$\"1q8#o;P%R'*F,7$$\"1nmmT9C#e\"Fgt$\"1k5 ecahy'*F,7$$\"1++D1*3`i\"Fgt$\"1n6j-RA:(*F,7$$\"1MLL$*zym;Fgt$\"1s^6-V lY(*F,7$$\"1ML$3N1#4Fgt$\"1.V%*zqlv)*F,7$ $\"1++v.Uac>Fgt$\"1sv$eCa$*))*F,7$$\"#?F($\"1'Q2&39V-**F,-%'COLOURG6&% $RGBG$\"#5!\"\"F(F(-%&TITLEG6#Q3CDF~for~Gamma(2,3)6\"-%+AXESLABELSG6$Q \"xFh\\l%!G-%%VIEWG6$;F(Fi[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 29 "f: =x->GammaPDF(alpha,beta,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR 6#%\"xG6\"6$%)operatorG%&arrowGF(-%)GammaPDFG6%%&alphaG%%betaG9$F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "F:=x->GammaCDF(alpha,be ta,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6#%\"xG6\"6$%)operato rG%&arrowGF(-%)GammaCDFG6%%&alphaG%%betaG9$F(F(F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "a:=plot(f(x),x=0..20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "b:=plot(f(x),x=0..5,color=yellow,filled=true) :" }}}{EXCHG {PARA 0 "> " 0 "" 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(7&FgbmFbbm7$$\"1,++D>#[Z%FV$\"1l^f(pe(=6F,7$FjbmF(7&F^cmFibm7$$\"1nmT &G!e&e%FV$\"1&p&[;^*[5\"F,7$FacmF(7&FecmF`cm7$$\"1MLL$)Qk%o%FV$\"1>(yw ++@4\"F,7$FhcmF(7&F\\dmFgcm7$$\"1+]iSjE!z%FV$\"1US'\\8!4y5F,7$F_dmF(7& FcdmF^dm7$$\"1,]P40O\"*[FV$\"1='Qk1kV1\"F,7$FfdmF(7&FjdmFedm7$$\"\"&F( $\"1AJl/*3$\\5F,7$F]emF(7\"-%&STYLEG6#%,PATCHNOGRIDG-F^`l6&F``l$\"*+++ +\"!\")FiemF(-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fi_l%(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 -1 103 "Find the value of the yellow shaded area, the area under the Gamma(2,3) density curve t o the left of 5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(F(5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+esJj\\!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 74 "The value of this area can also be found by using \+ the integration command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(int(f(x),x=0..5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+esJj\\!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a:=plot(f(x),x=0..20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "b:=plot(f(x),x=6..10,color=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 -1 103 "F ind the value of the yellow shaded area, the area under the Gamma(2,3) density curve between 6 and 10." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7^o7 $\"\"!F(7$$\"1LLL3x&)*3\"!#;$\"1UX!=H]x;\"!#<7$$\"1nmm;arz@F,$\"1[Q2.y <_AF/7$$\"1+++DJdpKF,$\"1R!4taTxD$F/7$$\"1LLLL3VfVF,$\"15g$zU)o)=%F/7$ $\"1mm\"z>5xI&F,$\"1@E)fNY6%\\F/7$$\"1****\\i&*)fD'F,$\"1O\\.ZOsUcF/7$ $\"1LL3F*oU?(F,$\"1zR!R6eeH'F/7$$\"1nmm\"H[D:)F,$\"1)[@T7&)G!pF/7$$\"1 ++v$pU&G5!#:$\"12C*Gvg66)F/7$$\"1LLLe0$=C\"FV$\"1yX5?Q3@\"*F/7$$\"1LLL 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F(7&FbcmF]cm7$$\"1******Rv&)z&*FV$\"1C#fhKv!oVF/7$FecmF(7&FicmFdcm7$$ \"1LLLGUYo'*FV$\"17F'Q1u,G%F/7$F\\dmF(7&F`dmF[dm7$$\"1mmm1^rZ(*FV$\"1T ggBXv-UF/7$FcdmF(7&FgdmFbdm7$$\"1++]sI@K)*FV$\"1R3zN5X@TF/7$FjdmF(7&F^ emFidm7$$\"1++]2%)38**FV$\"1grr\")o#[/%F/7$FaemF(7&FeemF`em7$$Fb`lF($ \"1)f$eQqxjRF/7$FhemF(7\"-%&STYLEG6#%,PATCHNOGRIDG-F^`l6&F``l$\"*++++ \"!\")FcfmF(-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fi_l%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Using the CDF, we \+ find the area to the left of 10 and then subtract the area to the left of 6." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "area1:=F(10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&a rea1G$\"+bp7a%)!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "area2 :=F(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&area2G$\"+/:%*Rf!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "area_between:=area1-area2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-area_betweenG$\"+^a=9D!#5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Us ing the integration command, we simply integrate the GammaPDF from 6 t o 10." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(int(f(x),x=6..10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^a=9D!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "a:=plot(f(x),x=0..20,labels=[\" \",\" \"]):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 64 "b:=plot(f(x),x=8..20,color=yellow,filled=true, labels=[\" \",\" \"]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "c :=textplot([8,-.01,\"x\"],color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "d:=textplot([10,0.02,\".25\"],color=black):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display([a,b,c,d]);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Find the value of " }{TEXT 286 1 "x" }{TEXT -1 3 ". " }{TEXT 287 1 " x" }{TEXT -1 65 " is known as the third quartile for the Gamma(2, 3) d istribution." }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6( -%'CURVESG6$7^o7$\"\"!F(7$$\"1LLL3x&)*3\"!#;$\"1UX!=H]x;\"!#<7$$\"1nmm ;arz@F,$\"1[Q2.y<_AF/7$$\"1+++DJdpKF,$\"1R!4taTxD$F/7$$\"1LLLL3VfVF,$ \"15g$zU)o)=%F/7$$\"1mm\"z>5xI&F,$\"1@E)fNY6%\\F/7$$\"1****\\i&*)fD'F, $\"1O\\.ZOsUcF/7$$\"1LL3F*oU?(F,$\"1zR!R6eeH'F/7$$\"1nmm\"H[D:)F,$\"1) [@T7&)G!pF/7$$\"1++v$pU&G5!#:$\"12C*Gvg66)F/7$$\"1LLLe0$=C\"FV$\"1yX5? Q3@\"*F/7$$\"1LLLLA`c9FV$\"1?[v!fl\"f**F/7$$\"1MLL3RBr;FV$\"1hXc&>(zj5 F,7$$\"1++vV^\"\\)=FV$\"1P&oL$fK<6F,7$$\"1nm;zjf)4#FV$\"1ktUnkYe6F,7$$ \"1++]Piq'H#FV$\"1ztqG&3o=\"F,7$$\"1LL$e4;[\\#FV$\"1`hO8vz17F,7$$\"1++ ]P=97F,7$$\"1nm;zt%**p#FV$\"1[v(=g1(>7F,7$$\"1ML$3-8D !GFV$\"19rQ(G)[B7F,7$$\"1++]i'y]!HFV$\"1Y(fN)yjD7F,7$$\"1ML$e>_6,$FV$ \"1t['>NciA\"F,7$$\"1nm;HdA7F,7$$\"1nm;/&R3a$FV$\"1X*H*z!y& 37F,7$$\"1++]7iI_PFV$\"1bFDOie$>\"F,7$$\"1nmm;_M(=%FV$\"12^au$o@:\"F,7 $$\"1MLL3y_qXFV$\"1#)R29x!o5\"F,7$$\"1,++]1!>+&FV$\"1C$3n%H/\\5F,7$$\" 1+++]Z/NaFV$\"1,R%*yxOm)*F/7$$\"1+++]$fC&eFV$\"1owmm67W#*F/7$$\"1ML$ez 6:B'FV$\"1!ejDcaXn)F/7$$\"1nmm;=C#o'FV$\"1&\\-vY]V+)F/7$$\"1nmmm#pS1(F V$\"1p/%>.q/X(F/7$$\"1++]i`A3vFV$\"1_rt<(o\"HoF/7$$\"1mmmm(y8!zFV$\"1e ^XP`./jF/7$$\"1,+]i.tK$)FV$\"1Nn#e9Eyv&F/7$$\"1,+](3zMu)FV$\"1CM1f$z&o _F/7$$\"1omm\"H_?<*FV$\"1_'HYE!3\"z%F/7$$\"1nm;zihl&*FV$\"1aV%f%eL#Q%F /7$$\"1LLL3#G,***FV$\"1d=^ge\"H(RF/7$$\"1LLezw5V5!#9$\"1c=w\\9C\"e$F/7 $$\"1++v$Q#\\\"3\"Ffx$\"1S'\\Ib%3nKF/7$$\"1LL$e\"*[H7\"Ffx$\"1$>**>P*[ aHF/7$$\"1+++qvxl6Ffx$\"1BI4,h6fEF/7$$\"1++]_qn27Ffx$\"1>DyVKh&R#F/7$$ \"1++Dcp@[7Ffx$\"1y='zSfI;#F/7$$\"1++]2'HKH\"Ffx$\"1Sgv/#>)G>F/7$$\"1n mmwanL8Ffx$\"1Vn#fnf#Q&o-9F/7$$\"1+++&)Hxe9Ffx$\"15J,l8,`7F/7$$\"1nm\"H!o -*\\\"Ffx$\"1l@M#44f7\"F/7$$\"1++DTO5T:Ffx$\"1Ej#[^Ng+\"F/7$$\"1nmmT9C #e\"Ffx$\"1@LQ$eh`+*!#=7$$\"1++D1*3`i\"Ffx$\"17eq!eFfx$\"1(o)[9nn$e$Fj\\l7$$\"1++v.Uac>Ffx$\"1^R?Y!fy>$Fj\\l 7$$\"#?F($\"1l%)39^2GGFj\\l-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%)POLYGONS G6U7&7$$\"\")F(F(7$Fi`l$\"1Q,OvnIwhF/7$$\"1+++]ech#)FV$\"1:@!=**)oXeF/ 7$F_alF(7&FcalF^al7$$\"1++](*G:*[)FV$\"1_xEBd*yc&F/7$FfalF(7&FjalFeal7 $$\"1+++N$)4X()FV$\"1H$RCg6nE&F/7$F]blF(7&FablF\\bl7$$\"1*****\\MSF+*F V$\"1COZIu*FV$\"1N6ccKG2UF/7$FiclF(7&F]dlFhcl7$$\"1++]xOi(***F V$\"1eFG_`(f'RF/7$F`dlF(7&FddlF_dl7$$\"1++vs$Q^-\"Ffx$\"19AE`G!ot$F/7$ FgdlF(7&F[elFfdl7$$\"1+++82C^5Ffx$\"1ievqOj7NF/7$F^elF(7&FbelF]el7$$\" 1++]o;Bu5Ffx$\"1gRve\"\\YK$F/7$FeelF(7&FielFdel7$$\"1+++RS6+6Ffx$\"1\\ bL(yBL7$F/7$F\\flF(7&F`flF[fl7$$\"1+++&o-h7\"Ffx$\"1ND&[(*eC6F/7$F\\]mF(7&F`]mF[]m7$$\"1++]JigC:Ffx$\"1KTwV)H:0\"F/7$ Fc]mF(7&Fg]mFb]m7$$\"1++vt,$*[:Ffx$\"1'RdcCm5&)*Fj\\l7$Fj]mF(7&F^^mFi] m7$$\"1++]kx$fd\"Ffx$\"1SI`'>Z*f\"*Fj\\l7$Fa^mF(7&Fe^mF`^m7$$\"1+++'G0 -g\"Ffx$\"1*QlKX^#y&)Fj\\l7$Fh^mF(7&F\\_mFg^m7$$\"1+++Xg6E;Ffx$\"1!yt* zb&e*zFj\\l7$F__mF(7&Fc_mF^_m7$$\"1++vV]f\\;Ffx$\"1V!RsY$p+vFj\\l7$Ff_ mF(7&Fj_mFe_m7$$\"1+++\"zj_n\"Ffx$\"1t'*\\+$\\F*pFj\\l7$F]`mF(7&Fa`mF \\`m7$$\"1++v\"3;%*p\"Ffx$\"1H@\"oYh[a'Fj\\l7$Fd`mF(7&Fh`mFc`m7$$\"1++ v%=iYs\"Ffx$\"1zC(o+2g5'Fj\\l7$F[amF(7&F_amFj`m7$$\"1+++l[M\\]c\\Fj\\l7$F`bmF(7&FdbmF_bm 7$$\"1++]5Q_D=Ffx$\"1:qA*oIxh%Fj\\l7$FgbmF(7&F[cmFfbm7$$\"1++vxSw]=Ffx $\"1)[8[S5QI%Fj\\l7$F^cmF(7&FbcmF]cm7$$\"1+++is&R(=Ffx$\"1J(f#R#QN.%Fj \\l7$FecmF(7&FicmFdcm7$$\"1++]o#R0!>Ffx$\"18v`S-)Qu$Fj\\l7$F\\dmF(7&F` dmF[dm7$$\"1+++KXJC>Ffx$\"1?AccN*=]$Fj\\l7$FcdmF(7&FgdmFbdm7$$\"1++v@R m\\>Ffx$\"1Zh.\"yS0E$Fj\\l7$FjdmF(7&F^emFidm7$$\"1++DAl#R(>Ffx$\"1&>K] V[Y/$Fj\\l7$FaemF(7&FeemF`emFh_l7$Fi_lF(7\"-%&STYLEG6#%,PATCHNOGRIDG-F ^`l6&F``l$\"*++++\"!\")F_fmF(-%%TEXTG6%7$Fi`l$Fc`l!\"#%\"xG-F^`l6&F``l F(F(F_fm-Fcfm6%7$$Fb`lF($\"\"#Fgfm%$.25G-F^`l6&F``lF(F(F(-%+AXESLABELS G6$Q\"~6\"Fggm-%%VIEWG6$;F(Fi_l%(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 18 "Find the value of " }{TEXT 285 3 "x, \+ " }{TEXT -1 138 "which has .75 area to the left of it. Maple requires a search range of x values in order to solve the equation. We chose \+ to use x=0..20." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "fsolve(F(x)=.75,x=0..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(e.z2)!\"*" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Properties of the Distribution" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 281 61 "Calculation o f Mean, Variance, and Moment Generating Function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "To calculate the mean of \+ the Gamma distribution, we can simply integrate " }{TEXT 282 1 "x" } {TEXT -1 62 "*(Gamma PDF), per the definition of mathematical expecta tion." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "with(plots, display):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(showassumed=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "assume(alpha > 0); assume(beta > 0);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "f:=x->GammaPDF(alpha,beta,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%)GammaPDFG6%% &alphaG%%betaG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "E X:=int(x*f(x),x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EX G*&%&beta|irG\"\"\"%'alpha|irGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Calculating Var(" }{TEXT 256 1 "X " }{TEXT -1 29 "), the variance of the Gamma(" }{XPPEDIT 18 0 "alpha,b eta;" "6$%&alphaG%%betaG" }{TEXT -1 16 " ) distribution." }}{PARA 0 " " 0 "" {TEXT -1 33 "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*()%&beta|irG\"\"#\"\"\"%'alpha|irG\" \"\",&F*F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "VarX:=s implify(E_X_SQ-EX^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VarXG*&)%& beta|irG\"\"#\"\"\"%'alpha|irG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The Moment Generating Function (MGF) can be easily calculated via Maple. Recall the moment generati ng function of a random variable " }{TEXT 274 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#-%$expG6#%#tXG" } {TEXT -1 63 "), provided this expectation exists. In taking the integ ral of" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(tx)" "6#- %$expG6#%#txG" }{TEXT -1 31 "*GammaPDF(x) over the range (0," } {XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 43 "), the argument of the exponential term is " }{XPPEDIT 18 0 "x(t-1/beta);" "6#-%\"xG6 #,&%\"tG\"\"\"*&\"\"\"F(%%betaG!\"\"F," }{TEXT -1 64 ", and for the re sulting integral to be convergent, we must have " }{XPPEDIT 18 0 "x(t- 1/beta);" "6#-%\"xG6#,&%\"tG\"\"\"*&\"\"\"F(%%betaG!\"\"F," }{TEXT -1 30 "<0. Equivalently, we require " }{XPPEDIT 18 0 "t < 1/beta;" "6#2% \"tG*&\"\"\"\"\"\"%%betaG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "assume(t<1/beta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "int(exp(t*x)*f(x),x=0..infinity);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&\"\"\"F$*&)%&beta|irG%'alpha|irG\"\"\"),$*&,&*&%#t| irG\"\"\"F'F0F0!\"\"F0F$F'!\"\"F1F(\"\"\"F2" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "So the moment generatin g function for a Gamma( " }{XPPEDIT 18 0 "alpha,beta;" "6$%&alphaG%%be taG" }{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-beta*t)^(-alpha);" "6#),&\"\"\"\"\"\"*&%%betaGF&%\"tGF&!\"\",$%&alph aGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 15 "We will define " }{TEXT 262 1 "M" }{TEXT -1 2 "( " } {TEXT 263 1 "t" }{TEXT -1 20 " ) as a function of " }{TEXT 264 1 "t" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "M(t):=(1-beta*t)^(-alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"MG6#%\"tG),&\"\"\"F**&%%betaGF* F'F*!\"\",$%&alphaGF-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 122 "The moment generating function provides \+ us with alternative 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 -1 209 ") 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. Looking at the first derivative, w e find:" }}{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#>%$M_pG*&*(),&\"\"\"F)*&%%betaGF)%\"tGF)!\"\",$%&alphaGF-F)F/F)F +\"\"\"F0F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplif y(M_p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*(),&\"\"\"F(*&%%betaGF (%\"tGF(!\"\",$%&alphaGF,F(F.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#*&%&alphaG\"\"\"%%betaGF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "This " }{TEXT -1 35 "agrees with our previous result, E(" }{TEXT 272 1 "X" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "alpha*beta;" "6#*&%&alphaG\"\"\"%%betaGF% " }{TEXT -1 45 ". Turning to the second derivative, we find:" }} {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**&%%betaGF*%\"tGF*!\"\",$%&alphaGF.F*)F0\"\" #\"\"\")F,F2F3F3*$)F)\"\"#F3!\"\"F**&*(F(F3F0F*F4F3F3*$)F)\"\"#F3F8F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(subs(t=0,M_pp) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)%&alphaG\"\"#\"\"\")%%betaG F'F(\"\"\"*&F&F+F)F(F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "Therefore, E(" }{XPPEDIT 18 0 "X^2;" "6#* $%\"XG\"\"#" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "alpha^2*beta^2+alpha*b eta^2" "6#,&*&%&alphaG\"\"#%%betaG\"\"#\"\"\"*&F%F)*$F'\"\"#F)F)" } {TEXT -1 109 ", which again is in agreement with the value calculated \+ previously. The variance is now quickly calculated as" }}{PARA 0 "" 0 "" {TEXT -1 4 "Var(" }{TEXT 273 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 "alpha^2*beta^2+alpha*beta^2-(alpha*beta)^2;" "6#,(*&%&alphaG\"\"#%%betaG\"\"#\"\"\"*&F%F)*$F'\"\"#F)F)*$*&F%F)F'F) \"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 14 " = " }{XPPEDIT 18 0 "alpha*beta^2;" "6#*&%&alphaG\"\"\"*$%%betaG\"\"#F%" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Simulation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 278 "We will now simulate som e data from an Gamma distribution in order to compare the sampling dis tribution to the theoretical distribution. The next bit of code will \+ simulate a random sample of size 500 from a Gamma(2,3) distribution, b ut you are invited to change the parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plo ts):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "alpha:=2: beta:=3: \+ n:=500:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sample:=GammaS(a lpha,beta,n):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "Remember that when you plot histograms, the number of bins that you select can make a huge difference in the plot." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "num_bins:=26:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a:=H istogram(sample,0..26,num_bins):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "b:=plot(GammaPDF(alpha,beta,x),x=0..26,color=black): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display([a,b],title=\"S imulated vs. Theoretical\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7dq7$\"\"!F(7$F($\"1+++++++]!#<7$$\"\" \"F(F*7$F.F(F07$F.$\"1+++++++#*F,7$$\"\"#F(F27$F5F(F77$F5$\"1++++++?7! #;7$$\"\"$F(F97$F=F(F?7$F=$\"1++++++g8F;7$$\"\"%F(FA7$FDF(FF7$FD$\"1++ +++++yF,7$$\"\"&F(FH7$FKF(FM7$FK$\"1++++++S6F;7$$\"\"'F(FO7$FRF(FT7$FR FH7$$\"\"(F(FH7$FWF(FY7$FW$\"1+++++++cF,7$$\"\")F(Fen7$FhnF(Fjn7$Fhn$ \"1+++++++#)F,7$$\"\"*F(F\\o7$F_oF(Fao7$F_o$\"1+++++++OF,7$$\"#5F(Fco7 $FfoF(Fho7$Ffo$\"1+++++++GF,7$$\"#6F(Fjo7$F]pF(F_p7$F]p$\"1+++++++SF,7 $$\"#7F(Fap7$FdpF(Ffp7$FdpFjo7$$\"#8F(Fjo7$FipF(F[q7$Fip$\"1+++++++9F, 7$$\"#9F(F]q7$F`qF(Fbq7$F`q$\"1+++++++!)!#=7$$\"#:F(Fdq7$FhqF(Fjq7$Fhq $FbpFfq7$$\"#;F(F\\r7$F^rF(F`r7$F^r$\"1+++++++7F,7$$\"#F(F\\r7$F asF(Fcs7$FasFir7$$\"#?F(Fir7$FfsF(Fhs7$Ffs$\"1+++++++?Ffq7$$\"#@F(Fjs7 $F]tF(F_tF\\t7$$\"#AF(Fjs7$FatF(FctFct7$$\"#BF(F(FdtFdtFdt7$$\"#CF(F(F gtFgt7$FhtFjs7$$\"#DF(Fjs7$F\\uF(F^uF^u7$$\"#EF(F(F_u-%'COLOURG6&%$RGB G$\"*++++\"!\")F(F(-%*LINESTYLEG6#F>-F$6$7boF'7$$\"1mm;/^2%3(F,$\"1dlE rZ](o(Ffq7$$\"1LL$3-:oT\"F;$\"1l.*)[.i,:F,7$$\"1++DJDAD@F;$\"1Pw\">jl) *>#F,7$$\"1nmmT+jLGF;$\"1%*3ogKqkGF,7$$\"1ML3_v.UNF;$\"1D?dFAJ(\\$F,7$ $\"1++]i]W]UF;$\"1?*[5pM))4%F,7$$\"1nm\"Hd_)e\\F;$\"1NUs_qPqYF,7$$\"1L LL$3gsm&F;$\"1fCC/5,8_F,7$$\"1l;HdK-+pF;$\"1B!*R&4S94'F,7$$\"1***\\7V' yK\")F;$\"1\"Fiy$\"1*y&o$\\^2$*)F,7$$\"1+](=]0 rL\"Fiy$\"1Z()RLB(Q^*F,7$$\"1mT&QhUdZ\"Fiy$\"1!p2S/6E+\"F;7$$\"1LL$esz Vh\"Fiy$\"15n`yyEZ5F;7$$\"1MLL.>\\$*=Fiy$\"1YoE.d?>6F;7$$\"1ML$33/E<#F iy$\"1H4XB7F;7$$\"1LL$3qYp& GFiy$\"1%Q3$Qb#[A\"F;7$$\"1m;z/CL@HFiy$\"1\"fqxvNeA\"F;7$$\"1++v3\"=d) HFiy$\"1da$R'3DE7F;7$$\"1mmm;&*[9JFiy$\"1t-S=URD7F;7$$\"1LLeC4EVKFiy$ \"1n-TU[WA7F;7$$\"1mm\"HfJ*4NFiy$\"1%Ho53N/@\"F;7$$\"1***\\7E-mx$Fiy$ \"1h#R\"))[i\">\"F;7$$\"1LLeMY=GVFiy$\"1!ecEc.j8\"F;7$$\"1++Dw!)*z([Fi y$\"1@\"**\\\"[>m5F;7$$\"1mmm\"y[NW&Fiy$\"1/\"Q()GLQ&)*F,7$$\"1LL$3:'o TfFiy$\"1Q%[UlM+6*F,7$$\"1+++X3Z-lFiy$\"1)p1_CD+F)F,7$$\"1+++v\"eb1(Fi y$\"1)*)R?$GM[uF,7$$\"1+++br>3wFiy$\"1jNL)*RH$p'F,7$$\"1KLeM`'45)Fiy$ \"1+*zezqs/'F,7$$\"1mmmhV\"po)Fiy$\"1_FGns7M`F,7$$\"1mmmY+H$=*Fiy$\"1L 3w1],zZF,7$$\"1)**\\7(Hpg(*Fiy$\"1R;laT=!>%F,7$$\"1nmmR#zr-\"!#9$\"1zl QOp&)=PF,7$$\"1+]7Z\\D$3\"Ffbl$\"1()R;h$RKD$F,7$$\"1+]P\"G_m8\"Ffbl$\" 1OJ))[U,dGF,7$$\"1nm\"zzmB>\"Ffbl$\"1Hd\\NW3*[#F,7$$\"1n;H;,`V7Ffbl$\" 128z.!o))=#F,7$$\"1LL3nmr)H\"Ffbl$\"1dk6to*=!>F,7$$\"1L$eM)*RgN\"Ffbl$ \"1\"=T_iU/k\"F,7$$\"1+]())4SfS\"Ffbl$\"1f1HvD=S9F,7$$\"1LLe!fL)f9Ffbl $\"1q#R59(\\\\7F,7$$\"1+++T3^::Ffbl$\"18r9S?Vx5F,7$$\"1++Do,)*p:Ffbl$ \"1kDP\"GZ$3$*Ffq7$$\"1+]7V?oA;Ffbl$\"1#\\(4ou#32)Ffq7$$\"1++v*[)>\"o \"Ffbl$\"19P\"oii+)oFfq7$$\"1mmm>\"yPt\"Ffbl$\"1z8zAfcafFfq7$$\"1++](4 =**y\"Ffbl$\"1)G'paFA)4&Ffq7$$\"1L$e9E*yS=Ffbl$\"1Xdu]^LDWFfq7$$\"1++] !)[S'*=Ffbl$\"1d;#Gbvvy$Ffq7$$\"1m;zV[t[>Ffbl$\"1\")=v:b4pKFfq7$$\"1+] iLZV.?Ffbl$\"1WXznBo+GFfq7$$\"1mm;uQ\"p0#Ffbl$\"1KPqsa%fS#Ffq7$$\"1+]7 y:!H6#Ffbl$\"1ZdRdDo]?Ffq7$$\"1LLL\"RCo;#Ffbl$\"1dE;x)Rqv\"Ffq7$$\"1LL 3c#o>A#Ffbl$\"1)G:q0D#*\\\"Ffq7$$\"1n;z,blwAFfbl$\"1Sw')fF9!G\"Ffq7$$ \"1+++,u!pK#Ffbl$\"1p'Q-B1m5\"Ffq7$$\"1mmT[<]%Q#Ffbl$\"1Bydwg7f$*!#>7$ $\"1LLL>[,OCFfbl$\"1PO^o-w_!)Ffjl7$$\"1+]7(\\Q4\\#Ffbl$\"1$*>$z**Rn&oF fjl7$$\"1+]([Y2Na#Ffbl$\"10)*)fIhg(eFfjl7$F`u$\"1 " 0 "" {MPLTEXT 1 0 22 "theo_mean:=alpha*beta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*theo_meanG\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sim_mean:=Mean(sample);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)sim_mea nG$\"+'o=z(f!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "relativ e_error:=(sim_mean-theo_mean)/theo_mean;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/relative_errorG$!++!>-o$!#7" }}}}}{MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 }