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"Probability Distribution \+ Function and Shape" }}{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 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " " } {TEXT 256 35 " The Negative Binomial Distribution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Consider observing indepe ndent Bernoulli trials having success probability " }{TEXT 286 1 "p" } {TEXT -1 8 " until " }{TEXT 302 1 "r" }{TEXT -1 10 " successes" }} {PARA 0 "" 0 "" {TEXT -1 56 "have been observed. For example, flippin g a coin until " }{TEXT 303 1 "r" }{TEXT -1 22 " heads are observed is " }}{PARA 0 "" 0 "" {TEXT -1 50 "such an exercise. If we define a ran dom variable " }{TEXT 288 1 "X" }{TEXT -1 44 " as the number of trials required to observe" }}{PARA 0 "" 0 "" {TEXT -1 10 "the first " } {TEXT 304 2 "r " }{TEXT -1 15 "successes then " }{TEXT 289 1 "X" } {TEXT -1 40 " is said to have the Negative Binomial (" }{TEXT 305 2 "r ," }{TEXT -1 1 " " }{TEXT 287 1 "p" }{TEXT -1 15 ") distribution." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 257 1 " X" }{TEXT -1 52 " is a random variable having the Negative Binomial ( " }{TEXT 306 5 "r , p" }{TEXT -1 36 ") distribution, then the probabil ity" }}{PARA 0 "" 0 "" {TEXT -1 26 "distribution function for " } {TEXT 259 1 "X" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 " " }{TEXT 260 1 "f" }{TEXT -1 3 " ( " }{TEXT 261 1 "k" }{TEXT -1 8 " ) = P( " }{TEXT 262 1 "X" } {TEXT -1 3 " = " }{TEXT 263 1 "k" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 " binomial(k-1,r-1)*(1-p)^(k-r)*p^r;" "6#*(-%)binomialG6$,&%\"kG\"\"\"\" \"\"!\"\",&%\"rGF)\"\"\"F+F)),&\"\"\"F)%\"pGF+,&F(F)F-F+F))F2F-F)" } {TEXT -1 4 " , " }{TEXT 300 1 "k" }{TEXT -1 3 " = " }{TEXT 307 1 "r" }{TEXT -1 2 ", " }{TEXT 308 1 "r" }{TEXT -1 4 "+1, " }{TEXT 309 1 "r" }{TEXT -1 7 "+2, ..." }}{PARA 0 "" 0 "" {TEXT 258 3 " " }}{PARA 0 " " 0 "" {TEXT -1 95 "The form of this function can be seen by way of th e following argument. The probability of any" }}{PARA 0 "" 0 "" {TEXT -1 18 "fixed sequence of " }{TEXT 310 1 "r" }{TEXT -1 15 " succe sses and " }{TEXT 311 3 "k-r" }{TEXT -1 26 " failures has probability \+ " }{XPPEDIT 18 0 "(1-p)^(k-r)*p^r" "6#*&),&\"\"\"\"\"\"%\"pG!\"\",&%\" kGF'%\"rGF)F')F(F,F'" }{TEXT -1 17 " . How many such" }}{PARA 0 "" 0 "" {TEXT -1 13 "sequences of " }{TEXT 312 1 "r" }{TEXT -1 15 " success es and " }{TEXT 313 3 "k-r" }{TEXT -1 32 " failures have a success on \+ the " }{TEXT 314 1 "k" }{TEXT -1 18 "th trial? Namely," }}{PARA 0 "" 0 "" {TEXT -1 9 "binomial(" }{TEXT 315 1 "k" }{TEXT -1 3 "-1," }{TEXT 316 1 "r" }{TEXT -1 40 "-1), as we are free to assign the first " } {TEXT 317 1 "r" }{TEXT -1 29 "-1 successes among the first " }{TEXT 318 1 "k" }{TEXT -1 10 "-1 trials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The follo wing code will draw a probability histogram for the Negative Binomial \+ (" }{TEXT 319 1 "r" }{TEXT -1 2 ", " }{TEXT 290 1 "p" }{TEXT -1 14 ") \+ distribution" }}{PARA 0 "" 0 "" {TEXT -1 19 "for your choice of " } {TEXT 320 1 "r" }{TEXT -1 6 " and " }{TEXT 291 1 "p" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r:=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p:=0.5;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"pG$\"\"&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ProbHist(NegBinomialPDF(r,p,x),1..20,20);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7\\p7$$\"1 +++++++]!#;\"\"!F'7$$\"1+++++++:!#:F+F,F,F,7$$\"1+++++++DF/F+F0F0F07$$ \"1+++++++NF/F+F3F37$F4$\"1++++++]i!#<7$$\"1+++++++XF/F77$F;F+F=7$F;$ \"1++++++]7F*7$$\"1+++++++bF/F?7$FBF+FD7$FB$\"1+++++]i:F*7$$\"1+++++++ lF/FF7$FIF+FKFH7$$\"1+++++++vF/FF7$FMF+FO7$FM$\"1++++v=n8F*7$$\"1+++++ ++&)F/FQ7$FTF+FV7$FT$\"1+++++v$4\"F*7$$\"1+++++++&*F/FX7$FenF+Fgn7$Fen $\"1++++]7.#)F97$$\"1++++++]5!#9Fin7$F\\oF+F_o7$F\\o$\"1++++]PfeF97$$ \"1++++++]6F^oFao7$FdoF+Ffo7$Fdo$\"1+++8.KGSF97$$F@F^oFho7$F[pF+F\\p7$ F[p$\"1+++voa&o#F97$$\"1++++++]8F^oF^p7$FapF+Fcp7$Fap$\"1+++pagXF^oFir7$F\\sF +F^s7$F\\s$\"1+++DT5T#*!#>7$$\"1++++++]?F^oF`s7$FdsF+-%+AXESLABELSG6$% !GFjs-%'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%%VIEWG6$%(DEFAULTGFet" 1 2 0 1 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "To see the effect of " }{TEXT 292 1 "p" }{TEXT -1 40 " on the shape of the Negative Binomial (" }{TEXT 321 1 " r" }{TEXT -1 2 ", " }{TEXT 293 1 "p" }{TEXT -1 29 ") distribution, the following" }}{PARA 0 "" 0 "" {TEXT -1 58 "animation will draw a serie s of probability histograms as " }{TEXT 294 1 "p" }{TEXT -1 27 " varie s from 0.05 to 0.5 by" }}{PARA 0 "" 0 "" {TEXT -1 46 "increments of 0. 05, assuming a fixed value of " }{TEXT 322 1 "r" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r:=4:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for n from 0 to 9 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "num:=convert(evalf(0. 05+n*0.05), string):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "tracker[n]: =textplot([30,0.12,`p is `.num],color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "H[n]:=ProbHist(NegBinomialPDF(r,0.05+n*0.05,x),1..40, 40):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "P[n]:=display(\{H[n],tracke r[n]\}):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 79 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zn&Fjr7$F\\\\mF\\\\`lF^\\mF^\\m7$F\\\\m$\"1+++)zfx\"RFjr7$Fd\\mF`\\`lF f\\mFf\\m7$Fd\\m$\"1+++'oe')o#Fjr7$F[]mFd\\`lF]]mF]]m7$F[]m$\"1+++-m_N =Fjr7$Fb]mFh\\`lFd]mFd]m7$Fb]m$\"1+++(ysnC\"Fjr7$Fi]mF\\]`lF[^mF[^m7$F i]m$\"1+++/0EF%)F`t7$Fa^mF`]`lFc^mFc^m7$Fa^m$\"1+++vzCpcF`t7$Fh^mFd]`l Fj^mFj^m7$Fh^m$\"1+++59P'z$F`t7$F__mFh]`lFa_mFa_m7$F__m$\"1+++uU\"4`#F `t7$Fg_mF\\^`lFi_mFi_m7$Fg_m$\"1+++TM+!o\"F`t7$F^`mF`^`lF``mF``m7$F^`m $\"1+++\\2^56F`t7$Fe`mFd^`lFg`mFg`m7$Fe`m$\"1+++JC'3J(Ffu7$F]amFh^`lF_ amF2-F(6%F*%)r~is~20.GF2F`amFdam-%&TITLEG6#Q " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Sa mple Probability Calculations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 31 "Sample Probability Calculations" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The follo wing code will draw a graph of the cumulative distribution function (C DF) for the " }}{PARA 0 "" 0 "" {TEXT -1 19 "Negative Binomial (" } {TEXT 360 1 "r" }{TEXT -1 2 ", " }{TEXT 358 1 "p" }{TEXT -1 35 ") dist ribution for your choice of " }{TEXT 359 1 "p" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r:=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p:=0.5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG $\"\"&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "PlotDiscCDF( NegBinomialPDF(r,p,x),1..20);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "69-%'CURVESG6#7$7$$\"#6\"\"!$\"1++++v=n))!#;7$$\"#7F* F+-F$6#7$7$$\"\"%F*$\"1++++++]i!#<7$$\"\"&F*F7-F$6#7$7$$\"\"\"F*F*7$$ \"\"#F*F*-F$6#7$FC7$$\"\"$F*F*-F$6#7$7$$\"\")F*$\"1++++v=njF-7$$\"\"*F *FR-F$6#7$7$F;$\"1++++++v=F-7$$\"\"'F*Fen-F$6#7$FI7$F5F*-F$6#7$7$FU$\" 1++++v$4Y(F-7$$\"#5F*Fbo-F$6#7$7$Feo$\"1+++++D\"G)F-7$F(F[p-F$6#7$7$Fh n$\"1+++++]PMF-7$$\"\"(F*Fbp-F$6#7$7$Fep$\"1+++++++]F-7$FPF[q-F$6#7$7$ $\"#8F*$\"1+++>UdQ&*F-7$$\"#9F*Fdq-F$6#7$7$F/$\"1+++J&>+F*F-7$FbqF]r-F $6#7$7$$\"#:F*$\"1+++](=U#)*F-7$$\"#;F*Ffr-F$6#7$7$Fgq$\"1+++mZ88(*F-7 $FdrF_s-F$6#7$7$$\"#=F*$\"1+++!z5B'**F-7$$\"#>F*Fhs-F$6#7$7$$\"# " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We can calculate probabil ities for the Negative Binomial (" }{TEXT 364 1 "r" }{TEXT -1 2 ", " } {TEXT 363 1 "p" }{TEXT -1 43 ") distribution using either the probabil ity" }}{PARA 0 "" 0 "" {TEXT -1 90 "distribution function (PDF) or the cumulative distribution function (CDF). We will denote" }}{PARA 0 " " 0 "" {TEXT -1 11 "the PDF by " }{TEXT 361 1 "f" }{TEXT -1 16 " and t he CDF by " }{TEXT 362 1 "F" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r:=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p:=0.5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG $\"\"&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=x->NegBin omialPDF(r,p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6 $%)operatorG%&arrowGF(-%/NegBinomialPDFG6%%\"rG%\"pG9$F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "F:=x->NegBinomialCDF(r,p,x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6#%\"xG6\"6$%)operatorG%&ar rowGF(-%/NegBinomialCDFG6%%\"rG%\"pG9$F(F(F(" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Below you will find thr ee different ways to calculate P(4<= " }{TEXT 323 1 "X" }{TEXT -1 7 " \+ <= 8)." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(f(4)+f(5)+f(6)+f(7)+f(8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")v=nj!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(sum(f(x),x=4..8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"++v=nj!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(F(8)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++v=nj!#5" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Properties of the Distribution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 61 "Calculation of M ean, Variance, and Moment Generating Function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Calculating E(" }{TEXT 266 1 "X" } {TEXT -1 53 "), the expectation or mean of the Negative Binomial (" } {TEXT 355 1 "r" }{TEXT -1 3 ", " }{TEXT 295 1 "p" }{TEXT -1 15 ") dis tribution." }}{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 28 "f:= x->NegBinomialPDF(r,p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6# %\"xG6\"6$%)operatorG%&arrowGF(-%/NegBinomialPDFG6%%\"rG%\"pG9$F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "EX:=simplify(sum(x*f(x) ,x=r..infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EXG*&%\"rG\"\" \"%\"pG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 16 "Calculating Var(" }{TEXT 267 1 "X" }{TEXT -1 42 "), the variance of the Negative Binomial (" }{TEXT 324 1 "r" }{TEXT -1 2 ", \+ " }{TEXT 296 1 "p" }{TEXT -1 15 ") distribution." }}{PARA 0 "" 0 "" {TEXT -1 33 "We will employ the formula: Var(" }{TEXT 268 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 46 "E_X_SQ:=simplify(sum(x^2*f(x),x=r..infinity));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'E_X_SQG*&*&%\"rG\"\"\",(F(F(%\"pG! \"\"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,$*&*&,&!\"\"\"\"\"%\"pGF*F*%\"rGF*\"\"\"*$)F+ \"\"#F-!\"\"F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The Moment Generating Function (MGF) can be easily c alculated via Maple." }}{PARA 0 "" 0 "" {TEXT -1 59 "Recall the moment generating function of a random variable " }{TEXT 301 1 "X" }{TEXT -1 11 " is defined" }}{PARA 0 "" 0 "" {TEXT -1 3 "as " }{TEXT 269 1 "M " }{TEXT -1 1 "(" }{TEXT 270 3 " t " }{TEXT -1 6 ") = E(" }{XPPEDIT 18 0 "exp(tX)" "6#-%$expG6#%#tXG" }{TEXT -1 36 "), provided this expec tation exists." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "simplify(sum(exp(t*x)*f(x), x=r..infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$expG6#*&%\"tG \"\"\"%\"rGF)F))%\"pGF*F)),(*&-F%6#F(F)F,F)F)F0!\"\"F)F),$F*F2F)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "So we have the moment generating function for a Negative Binomial (" } {TEXT 325 1 "r" }{TEXT -1 2 ", " }{TEXT 297 1 "p" }{TEXT -1 18 ") rand om variable " }}{PARA 0 "" 0 "" {TEXT -1 11 "is given by" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 271 1 "M" }{TEXT -1 2 "( " }{TEXT 272 1 "t" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "(exp(t)*p/(1-exp(t)+exp(t)*p))^r;" "6#)*(-%$expG6#%\"tG\"\"\"%\" pGF),(\"\"\"F)-F&6#F(!\"\"*&-F&6#F(F)F*F)F)F/%\"rG" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "We will d efine " }{TEXT 273 1 "M" }{TEXT -1 2 "( " }{TEXT 274 1 "t" }{TEXT -1 20 " ) as a function of " }{TEXT 275 1 "t" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "M:=t->(exp(t)*p/(1-exp(t)+exp(t)*p))^r;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"MGR6#%\"tG6\"6$%)operatorG%&arrowGF()*&*&-%$ expG6#9$\"\"\"%\"pGF3F3,(\"\"\"F6F/!\"\"*&F/F6F4F6F6!\"\"%\"rGF(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The moment generating function provides us alternative ways to calcul ate the" }}{PARA 0 "" 0 "" {TEXT -1 40 "mean and variance by way of th e 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 276 1 "t" } {TEXT -1 14 ") denotes the " }{TEXT 277 1 "r" }{TEXT -1 18 " th deriva tive of " }{TEXT 278 1 "M" }{TEXT -1 1 "(" }{TEXT 279 1 "t" }{TEXT -1 18 ") with respect to " }{TEXT 280 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 281 1 "M" }{TEXT -1 1 "(" }{TEXT 282 1 "t" }{TEXT -1 45 ") exists in an open interval containing zero." }}{PARA 0 "" 0 " " {TEXT -1 79 "See, for example, Mathematical Statistics and Data Anal ysis by John A. Rice for" }}{PARA 0 "" 0 "" {TEXT -1 39 "more on the m oment generating function." }}{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*&**)*&*&-%$expG6#%\"tG\"\"\"%\"pGF.F .,(*&F*\"\"\"F/F2F2F*!\"\"F2F2!\"\"%\"rGF2F5F2,&F(F2*&*(F*F.F/F.,&F*F3 F1F2F2F.*$)F0\"\"#F.F4F3F2F0F2F.*&F*\"\"\"F/\"\"\"F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(M_p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*(-%$expG6#,$%\"tG!\"\"\"\"\"%\"rGF+)*&%\"pG\"\"\",(F /F+F*F+F%F+!\"\"F,F+F0F1F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify(subs(t=0,M_p));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"r G\"\"\"%\"pG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 17 "And therefore if " }{TEXT 283 1 "X" }{TEXT -1 25 " is a Negative Binomial (" }{TEXT 326 1 "r" }{TEXT -1 2 ", " }{TEXT 298 1 "p" }{TEXT -1 19 ") variable, then E(" }{TEXT 284 1 "X" }{TEXT -1 4 ") = " }{TEXT 327 1 "r" }{TEXT -1 1 "/" }{TEXT 299 1 "p" }{TEXT -1 19 ", which agrees with" }}{PARA 0 "" 0 "" {TEXT -1 57 "what we fou nd 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 12 "" 1 "" {XPPMATH 20 "6#>%%M_ppG,**&**)*&*&-%$ expG6#%\"tG\"\"\"%\"pGF/F/,(*&F+\"\"\"F0F3F3F+!\"\"F3F3!\"\"%\"rGF3)F6 \"\"#F/),&F)F3*&*(F+F/F0F/,&F+F4F2F3F3F/*$)F1\"\"#F/F5F4F8F/)F1F8F/F/* &)F+\"\"#F/)F0\"\"#F/F5F3*&**F(F/F6F3,(F)F3F;!\"$*&*(F+F/F0F/)F=F8F/F/ *$)F1\"\"$F/F5F8F3F1F3F/*&F+\"\"\"F0\"\"\"F5F3*&**F(F/F6F/F:F3F1F/F/*& F+\"\"\"F0\"\"\"F5F4*&**F(F/F6F/F:F/F=F/F/*&F+\"\"\"F0\"\"\"F5F3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(subs(t=0,M_pp));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&%\"rG\"\"\",(F&F&%\"pG!\"\"F%F&F& \"\"\"*$)F(\"\"#F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Therefore E(" }{XPPEDIT 18 0 "X^2" "6#*$%\"XG\"\"#" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "r*(1-p+r)/(p^2);" "6#*(%\"rG\"\"\",(\"\"\"F%%\"pG !\"\"F$F%F%*$F(\"\"#F)" }{TEXT -1 45 " , which again is in agreement w ith the value" }}{PARA 0 "" 0 "" {TEXT -1 64 "calculated previously. T he variance is now quickly calculated as" }}{PARA 0 "" 0 "" {TEXT -1 4 "Var(" }{TEXT 285 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 "r*(1-p+r)/(p^2)-(r/p)^2;" "6#,&*(%\"rG\"\"\",(\"\"\"F&% \"pG!\"\"F%F&F&*$F)\"\"#F*F&*$*&F%F&F)F*\"\"#F*" }}{PARA 0 "" 0 "" {TEXT -1 14 " = " }{XPPEDIT 18 0 "r*(1-p)/(p^2);" "6#*(%\"r G\"\"\",&\"\"\"F%%\"pG!\"\"F%*$F(\"\"#F)" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 333 18 "Special Properties" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "You probably noticed that the probability histogram for a Negative Binomial (" }{TEXT 356 3 "r,p" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 60 "distribution looked close to Normal (i.e. bell-shaped) wh en " }{TEXT 357 1 "r" }{TEXT -1 24 " was large. Take a look" }}{PARA 0 "" 0 "" {TEXT -1 45 "again at the shape of the Negative Binomial (" }{TEXT 335 1 "r" }{TEXT -1 2 ", " }{TEXT 334 1 "p" }{TEXT -1 16 ") dis tribution, " }}{PARA 0 "" 0 "" {TEXT -1 4 " as " }{TEXT 336 1 "r" } {TEXT -1 48 " varies from 1 to 20, assuming a fixed value of " }{TEXT 337 1 "p" }{TEXT -1 30 ". A normal curve with the same" }}{PARA 0 "" 0 "" {TEXT -1 76 "mean and variance as the Negative Binomial will be o verlayed for comparison." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p:=0.5:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for r fr om 1 to 20 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "num:=convert(evalf (r), string):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "tracker[r]:=textpl ot([60,0.25,`r is `.num],color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "H[r]:=ProbHist(NegBinomialPDF(r,p,x),1..80,80):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "N[r]:=plot(NormalPDF(r/p,r*(1-p)/p^2,x),x = 1..8 0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "P[r]:=display(\{H[r],N[r],tr acker[r]\}):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "display([seq(P[r], r=1..20)], insequence=true, \+ title=\"Normal Approx. to the Negative Binomial. p=0.5 and r is increa sing\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%(A NIMATEG667'-%'CURVESG6$7jo7$$\"\"\"\"\"!$\"16'QtWcp>#!#;7$$\"1LeR(*oC: 7!#:$\"1-r#*4!=%=CF17$$\"1m;z%z$\\I9F5$\"1>)yJ+T7g#F17$$\"1+v=#pSdk\"F 5$\"1hoqg\\\"Qt#F17$$\"1LLe*e()4'=F5$\"1*\\KyY_t!GF17$$\"1#HKRJ*z9>F5$ \"15&zV8Le\"GF17$$\"1]7GQ5ho>F5$\"1-HkcJD?GF17$$\"13-jiFUA?F5$\"1$\\t2 O$f?GF17$$\"1n\"zp[Mi2#F5$\"1V4:qA&o\"GF17$$\"1$3xc$z&Q=#F5$\"1.'z.s3s z#F17$$\"1+]P%Q\"[\"H#F5$\"1u4ob=mhFF17$$\"1L3x\"GGn]#F5$\"1#p^Je^bk#F 17$$\"1mm;z^(>s#F5$\"1qG;c')GwCF17$$\"1v=n`.E4HF5$\"1OAz'R+UH#F17$$\"1 $3x\"Gba'4$F5$\"1sE#>z\\&)3#F17$$\"1\"H#o-2$QG$F5$\"1gtaM*y#o=F17$$\"1 +v=xe6rMF5$\"1Y8O)p%=U;F17$$\"13Fp^5SeOF5$\"1wkjAuN=9F17$$\"1EioXQF 5$\"1<#\\$=-u.7F17$$\"1DJq+9(H.%F5$\"1v)3N8PQ+\"F17$$\"1L$3_dc-A%F5$\" 1')zL35yD#)!#<7$$\"1vV[T$y3V%F5$\"1tE[X\\E;F`t7$$\"1mmm@BI`nF5$\"1e&>Um7$$\"1m;zH \"Qt<(F5$\"1Qs0bz9oMFeu7$$\"1nm\"z$RP,wF5$\"12ReY`D16Feu7$$\"1+D\"y\"[ TX%)F5$\"1+]j$fWsq)!#@7$$\"1K$3xpb%*G*F5$\"1zJA'HZ'*z%!#A7$$\"1nT&yN_a 3\"!#9$\"17,h3hYo')!#D7$$\"1+vorg]Z7F_w$\"1=@xT<4VM!#G7$$\"1nTNaA5:9F_ w$\"1hlvKM)*GE!#K7$$\"1+v$R&4;#e\"F_w$\"16^_!>&z9^!#P7$$\"1LLeg8+aF_w$\"14teUdG!\\(!#[7$$\"1++vc2vv?F_w$ \"1B>T)\\eex\"!#a7$$\"1++DwE%oC#F_w$\"10`B\"G*\\y\"*!#i7$$\"1++DV9s6CF _w$\"1W)>ys'G%=#!#p7$$\"1nTNfrWhDF_w$\"1U4L>37O!)!#x7$$\"1LLe!$0\"7$$\"1LL$GhW5A$F_w$\"1xJ)=K8CG#!$:\"7$$\"1+v=$\\G9R$F_ w$\"1G&pY<$3Ut!$F\"7$$\"1+DcRUn`NF_w$\"1J*>.j;)o@!$Q\"7$$\"1L$3_lgHs$F _w$\"1i)>&o_pw\\!$^\"7$$\"1L3FI%=%yQF_w$\"1aaSflY$[$!$j\"7$$\"1m;HU15Y SF_w$\"1fPWvn%e'p!$x\"7$$\"1nTNM`F?UF_w$\"1u!*yPcFp#*!$#>7$$\"1+D\"e\" \\*=P%F_w$\"1Vy!=bA#GI!$0#7$$\"1n;a<#[c`%F_w$\"1w'*>wDjmA!$?#7$$\"1*** *\\,9#[q%F_w$\"16>-NX[58!$O#7$$\"1**\\PdVKq[F_w$\"1lxM%p8&pU!$`#7$$\"1 *\\(=xpXI]F_w$\"1>uF)oWYH\"!$p#7$$\"1+]i\\pD3_F_w$\"1HOl8%ybJ\"!$)G7$$ \"1LL$G8=!o`F_w$\"1k+CTnGMH!$1$7$$\"1++D'H(fQbF_wF.7$$\"1mTNrt;$p&F_wF .7$$\"1++v!H`@'eF_wF.7$$\"1L3_re:@gF_wF.7$$\"1+v$HQft='F_wF.7$$\"1LLe% p`)\\jF_wF.7$$\"1*\\(oz,(*>lF_wF.7$$\"1nmmtD\"Qo'F_wF.7$$\"1m;z&3l8&oF _wF.7$$\"1L3_y,`F`t7$$\"1+++ ++++&*F5Fbhl7$FehlF.Fghl7$Fehl$\"1++++]il(*Feu7$$\"1++++++]5F_wFihl7$F \\ilF.F^il7$F\\il$\"1++++D\"G)[Feu7$$\"1++++++]6F_wF`il7$FcilF.Feil7$F cil$\"1+++]iSTCFeu7$$FielF_wFgil7$FjilF.F[jl7$Fjil$\"1+++DJq?7Feu7$$\" 1++++++]8F_wF]jl7$F`jlF.Fbjl7$F`jl$\"1+++Dc^.h!#?7$$\"1++++++]9F_wFdjl 7$FhjlF.Fjjl7$Fhjl$\"1+++8yv^IFfjl7$$\"1++++++]:F_wF\\[m7$F_[mF.Fa[m7$ F_[m$\"1+++2*ye_\"Ffjl7$$\"1++++++];F_wFc[m7$Ff[mF.Fh[m7$Ff[m$\"1+++IX RHwFev7$$\"1++++++]Fev7$$\"1++++++]>F_wF h\\m7$F[]mF.F]]m7$F[]m$\"1+++lJuO&*F[w7$$\"1++++++]?F_wF_]m7$Fb]mF.Fd] m7$Fb]m$\"1+++#er$oZF[w7$$\"1++++++]@F_wFf]m7$Fi]mF.F[^m7$Fi]m$\"1+++ \"z&=%Q#F[w7$$\"1++++++]AF_wF]^m7$F`^mF.Fb^m7$F`^m$\"1+++'*G4#>\"F[w7$ $\"1++++++]BF_wFd^m7$Fg^mF.Fi^m7$Fg^m$\"1+++![k/'f!#B7$$\"1++++++]CF_w F[_m7$F__mF.Fa_m7$F__m$\"1+++RAB!)HF]_m7$$\"1++++++]DF_wFc_m7$Ff_mF.Fh _m7$Ff_m$\"1+++?h6!\\\"F]_m7$$\"1++++++]EF_wFj_m7$F]`mF.F_`m7$F]`m$\"1 +++&f!e]u!#C7$$\"1++++++]FF_wFa`m7$Fe`mF.Fg`m7$Fe`m$\"1+++*H!HDPFc`m7$ $\"1++++++]GF_wFi`m7$F\\amF.F^am7$F\\am$\"1+++\\^ki=Fc`m7$$\"1++++++]H F_wF`am7$FcamF.Feam7$Fcam$\"1+++XdA8$*Fbw7$$\"1++++++]IF_wFgam7$FjamF. 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\"1WPsXi4\"H'Far7$F`[gl$\"1+%ea--&phFar7$F`^l$\"1#Qy\"oEiOfFar7$Fchhl$ \"1[&yvXFJl&Far7$Ff^l$\"1mN(3yyjI&Far7$F[ihl$\"1Tzp&R?Far7$F^`l$\"1^ d*y2&ys;Far7$Fa`_m$\"12^Cv,J<8Far7$Fd`l$\"1$G%>!3!3<5Far7$Fd^`m$\"1HEm qr$o#zF`t7$Fj`l$\"1)\\WqF%=!3'F`t7$F`al$\"1p0J#G'erKF`t7$Fcal$\"1?X5cS <_YV_bUjFfjl7$Fbbl$\"1/<+a;\"=D#Ffjl7$Febl$\"1\"Rm>0+o v(Fev7$Fhbl$\"1CVF/RPLCFev7$F[cl$\"15r*[\"=>\">(F[w7$F^cl$\"1&[*Row\\2 AF[w7$Facl$\"1gimY4'zI&F]_m7$Fdcl$\"1KK\\9xS!R\"F]_m7$Fgcl$\"1xwI`kd:J Fc`m7$Fjcl$\"1s " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Indeed the Negative Binomial ( " }{TEXT 338 3 "r,p" }{TEXT -1 35 ") distribution approaches a Normal( " }{XPPEDIT 18 0 "r/p" "6#*&%\"rG\"\"\"%\"pG!\"\"" }{TEXT -1 2 " ," } {XPPEDIT 18 0 "( r*(1-p))/p^2" "6#*(%\"rG\"\"\",&\"\"\"F%%\"pG!\"\"F%* $F(\"\"#F)" }{TEXT -1 2 ") " }}{PARA 0 "" 0 "" {TEXT -1 16 "distributi on as " }{TEXT 339 1 "r" }{TEXT -1 60 " grows large. This can be easi ly seen by realizing that if " }{TEXT 340 1 "X" }{TEXT -1 6 " is a " } }{PARA 0 "" 0 "" {TEXT -1 19 "Negative Binomial (" }{TEXT 341 3 "r,p" }{TEXT -1 17 ") variable, then " }{TEXT 342 1 "X" }{TEXT -1 29 " has t he same distribution as" }}{PARA 0 "" 0 "" {TEXT 343 1 "G" }{TEXT -1 1 "1" }{TEXT 345 4 " + G" }{TEXT -1 1 "2" }{TEXT 346 11 " + ... + Gr" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "where" }{TEXT 344 3 " Gi " }{TEXT -1 55 " are independent and identically distributed Geometric (" }{TEXT 347 1 "p" }{TEXT -1 19 ") random variables." }}{PARA 0 "" 0 "" {TEXT 348 1 "G" }{TEXT -1 84 "1 can be interpreted as the number of Bernoulli trials required to observe the first" }}{PARA 0 "" 0 "" {TEXT -1 9 "success, " }{TEXT 349 1 "G" }{TEXT -1 80 "2 the number of \+ Bernoulli trials following the first success required to observe" }} {PARA 0 "" 0 "" {TEXT -1 87 "the second success, etc. By the central l imit theorem, we know that the distribution of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 350 1 "G" }{TEXT -1 1 "1" }{TEXT 351 4 " + G" } {TEXT -1 1 "2" }{TEXT 352 12 " + ... + Gr " }{TEXT -1 24 "approaches n ormality as " }{TEXT 354 2 "r " }{TEXT -1 39 "grows large, and therefo re so does the " }}{PARA 0 "" 0 "" {TEXT -1 37 "distribution of a Nega tive Binomial (" }{TEXT 353 3 "r,p" }{TEXT -1 11 ") variable." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Simulation" }}{EXCHG {PARA 257 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 365 10 "Simulation" }} {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 237 "We will now simulate some data from a negative binomial distribut ion in order to compare the sampling distribution to the theoretical d istribution. The next bit of code will simulate a random sample of si ze 500 from a Negative Binomial(" }{TEXT 366 1 "r" }{TEXT -1 6 " = 4, \+ " }{TEXT 367 1 "p" }{TEXT -1 76 "= 0.3) distribution, but you are more than welcome to change the parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r:=4: p:=0.3: numsim:=5 00:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sample:=NegBinomialS (r,p,numsim):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a:=Histogr am(sample,0.5..30+0.5,30):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%Q3WARNIN G:~There~are6\"\"\")Q@~data~points~out~of~plot~range.F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "b:=ProbHist(NegBinomialPDF(r,p,x),1 ..30,30):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display([a,b], title=\"Simulated(red) vs. Theoretical(blue)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7dr7$$\"1+++++++]!#; \"\"!F'7$$\"1+++++++:!#:F+F,F,F,7$$\"1+++++++DF/F+F0F0F07$$\"1+++++++N F/F+F3F37$F4$\"1+++,83I\")!#=7$$\"1+++++++XF/F77$F;F+F=7$F;$\"1+++$Ho# H=!#<7$$\"1+++++++bF/F?7$FCF+FE7$FC$\"1+++V*3V*eFA7$$\"1+++++++lF/FG7$ FJF+FL7$FJ$\"1+++y[!y[&FA7$$\"1+++++++vF/FN7$FQF+FS7$FQ$\"1,++tqJ2nFA7 $$\"1+++++++&)F/FU7$FXF+FZ7$FX$\"1+++._K?vFA7$$\"1+++++++&*F/Ffn7$FinF +F[o7$Fin$\"1+++p0\"p0\"F*7$$\"1++++++]5!#9F]o7$F`oF+Fco7$F`o$\"1+++T] 1/lFA7$$\"1++++++]6FboFeo7$FhoF+Fjo7$Fho$\"1+++1\"p0\"pFA7$$\"1++++++] 7FboF\\p7$F_pF+Fap7$F_p$\"1,++rJ2(FA7$FfpF[zFhpFhp7$Ffp$\"1+++>; #Qa'FA7$F[qF_zF]qF]q7$F[q$\"1+++q,&*HeFA7$FbqFczFdqFdq7$Fbq$\"1+++(R17 5&FA7$FiqFgzF[rF[r7$Fiq$\"1+++7b)[R%FA7$F`rF[[lFbrFbr7$F`r$\"1+++&o_ct $FA7$FerF_[lFgrFgr7$Fer$\"1+++c#[z8$FA7$F\\sFc[lF^sF^s7$F\\s$\"1+++([> %3EFA7$FasFg[lFcsFcs7$Fas$\"1+++m,6[@FA7$FhsF[\\lFjsFjs7$Fhs$\"1+++p** GaeQw&F97$FjuFc]lF\\vF\\v7$ Fju$\"1+++C$Q!RXF97$F_vFg]lFavFav7$F_v$\"1+++ZggeNF97$FdvF[^lFfvFfv7$F dv$\"1+++1,XyFF97$FhvF_^lFjv-F\\w6&F^wF+F+F_w-%&TITLEG6#QESimulated(re d)~vs.~Theoretical(blue)6\"-%+AXESLABELSG6$%!GF\\_l-%%VIEWG6$%(DEFAULT GF`_l" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Now let's compare the simulated mean to t he theoretical mean." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "theo_mean:=r/p;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%*theo_meanG$\"+LLLL8!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sim_mean:=evalf(Mean(sample));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%)sim_meanG$\"+++?@8!\")" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "relative_error:=(sim_mean-theo_mean)/theo_mean;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%/relative_errorG$!+_(****4*!#7" }}}} }{MARK "0 0 0" 4 }{VIEWOPTS 1 1 0 1 1 1803 }