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{SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 43 "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 27 " The Geometric Distribution" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Consider observing independent Ber
noulli trials having success probability " }{TEXT 288 1 "p" }{TEXT -1 
16 " until the first" }}{PARA 0 "" 0 "" {TEXT -1 93 "success is observ
ed.  For example, flipping a coin until the first time a head is obser
ved is" }}{PARA 0 "" 0 "" {TEXT -1 50 "such an exercise.  If we define
 a random variable " }{TEXT 290 1 "X" }{TEXT -1 44 " as the number of \+
trials required to observe" }}{PARA 0 "" 0 "" {TEXT -1 23 "the first s
uccess then " }{TEXT 291 1 "X" }{TEXT -1 31 " is said to have the Geom
etric(" }{TEXT 289 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 43 " is a random variable having the Geometric(" }{TEXT 292 
1 "p" }{TEXT -1 49 ") distribution, then the probability distribution
" }}{PARA 0 "" 0 "" {TEXT -1 13 "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 "(1-p)^(k-1)*p" "6#
*&),&\"\"\"\"\"\"%\"pG!\"\",&%\"kGF'\"\"\"F)F'F(F'" }{TEXT -1 4 "  , \+
" }{TEXT 307 1 "k" }{TEXT -1 13 " = 1,2,3, ..." }}{PARA 0 "" 0 "" 
{TEXT 258 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 62 "This is readily seen \+
by noting that if Bernoulli trial number " }{TEXT 293 1 "k" }{TEXT -1 
41 " yields the first success, then the first" }}{PARA 0 "" 0 "" 
{TEXT 294 1 "k" }{TEXT -1 28 "-1 trials were all failures." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The following cod
e will draw a probability histogram for the Geometric(" }{TEXT 295 1 "
p" }{TEXT -1 14 ") distribution" }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }
{TEXT 296 2 "p=" }{TEXT -1 51 "0.5, but you are encouraged to try othe
r values of " }{TEXT 314 2 "p." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p:=0.5;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"pG$\"\"&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "ProbHist(GeometricPDF(p,x),1..15,15);" }}{PARA 13 "" 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "To see the effect of  " }{TEXT 
297 1 "p" }{TEXT -1 32 " on the shape of the Geometric( " }{TEXT 298 
1 "p" }{TEXT -1 29 ") distribution, the following" }}{PARA 0 "" 0 "" 
{TEXT -1 58 "animation will draw a series of probability histograms as
 " }{TEXT 299 1 "p" }{TEXT -1 27 " varies from 0.05 to 0.5 by" }}
{PARA 0 "" 0 "" {TEXT -1 18 "increments of 0.05" }}{PARA 0 "" 0 "" 
{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for n \+
from 0 to 9 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "num:=convert(eval
f(0.05+n*0.05), string):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "tracker
[n]:=textplot([12,0.4,`p is `.num],color=blue):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "H[n]:=ProbHist(GeometricPDF(0.05+n*0.05,x),1..15,15):
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "P[n]:=display(\{H[n],tracker[n]
\}):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "display(seq(P[n], n=0..9), insequence=true,title=\"p \+
is increasing\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 
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s$\"1+++pS,V5F]]m7$FcsF^emFesF2-F(6%F*%)p~is~.45GF2FfsFjs7&-F:6$7hnF=7
$F>F>7$FEF>FHFHF`al7$FMF_alFOFO7$FM$F_rF@7$FTF\\fmFVFV7$FT$\"1++++++]i
FC7$FenF_fmFgnFgn7$Fen$\"1++++++DJFC7$F\\oFcfmF^oF^o7$F\\o$\"1+++++]i:
FC7$FcoFgfmFeoFeo7$Fco$\"1+++++]7yFe`l7$FjoF[gmF\\pF\\p7$Fjo$\"1+++++D
1RFe`l7$FapF_gmFcpFcp7$Fap$\"1++++]7`>Fe`l7$FhpFcgmFjpFjp7$Fhp$\"1++++
]il(*F]]m7$F_qFggmFbqFbq7$F_q$\"1++++D\"G)[F]]m7$FgqF[hmFiqFiq7$Fgq$\"
1+++]iSTCF]]m7$F^rF_hmF`rF`r7$F^r$\"1+++DJq?7F]]m7$FerFchmFgrFgr7$Fer$
\"1+++Dc^.h!#?7$F\\sFghmF^sF^s7$F\\s$\"1+++8yv^IFihm7$FcsF\\imFesF2-F(
6%F*%)p~is~.50GF2FfsFjs-%&TITLEG6#Q0p~is~increasing6\"" 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 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Sample Prob
ability Calculations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 91 "The following code will draw a graph of the cum
ulative distribution function (CDF) for the " }}{PARA 0 "" 0 "" {TEXT 
-1 10 "Geometric(" }{TEXT 310 1 "p" }{TEXT -1 19 ") distribution for \+
" }{TEXT 311 2 "p=" }{TEXT -1 51 "0.5, but you are encouraged to try o
ther values of " }{TEXT 316 1 "p" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{MPLTEXT 0 21 0 "" }}}{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 37 "PlotDiscCDF(GeometricPDF(p,x),1..15
);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "64-%'CURVESG
6#7$7$$\"\"\"\"\"!$\"1+++++++]!#;7$$\"\"#F*F+-F$6#7$7$F/$\"1+++++++vF-
7$$\"\"$F*F5-F$6#7$7$F8$\"1++++++]()F-7$$\"\"%F*F>-F$6#7$7$FA$\"1+++++
+v$*F-7$$\"\"&F*FG-F$6#7$7$FJ$\"1+++++](o*F-7$$\"\"'F*FP-F$6#7$7$FS$\"
1+++++vV)*F-7$$\"\"(F*FY-F$6#7$7$Ffn$\"1++++](=#**F-7$$\"\")F*F\\o-F$6
#7$7$F_o$\"1++++v$4'**F-7$$\"\"*F*Feo-F$6#7$7$Fho$\"1+++](o/)**F-7$$\"
#5F*F^p-F$6#7$7$Fap$\"1+++vVB!***F-7$$\"#6F*Fgp-F$6#7$7$Fjp$\"1+++)=<^
***F-7$$\"#7F*F`q-F$6#7$7$$\"#:F*$\"1+++D[p****F-7$$\"#;F*F[r-F$6#7$7$
Fcq$\"1+++%fev***F-7$$\"#8F*Fdr-F$6#7$7$Fgr$\"1+++(Hz()***F-7$$\"#9F*F
]s-F$6#7$7$F`s$\"1+++\\'*Q****F-7$FiqFfs-%+AXESLABELSG6$%!GF\\t-%'COLO
URG6&%$RGBGF*F*$\"*++++\"!\")-%%VIEWG6$%(DEFAULTGFgt" 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 "" }{TEXT 264 
0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "We can calculate pr
obabilities for the Geometric( " }{TEXT 300 1 "p" }{TEXT -1 43 ") dist
ribution using either the probability" }}{PARA 0 "" 0 "" {TEXT -1 90 "
distribution function (PDF) or the cumulative distribution function (C
DF).  We will denote" }}{PARA 0 "" 0 "" {TEXT -1 11 "the PDF by " }
{TEXT 265 1 "f" }{TEXT -1 16 " and the CDF by " }{TEXT 266 1 "F" }
{TEXT -1 51 ".  We will consider these functions for a variable " }
{TEXT 301 1 "X" }{TEXT -1 7 " having" }}{PARA 0 "" 0 "" {TEXT -1 32 "t
he Geometric(0.5) distribution." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p:=0.5;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"pG$\"\"&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "f:=x->GeometricPDF(p,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%-GeometricPDF
G6$%\"pG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "F:=x->G
eometricCDF(p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6#%\"xG6\"
6$%)operatorG%&arrowGF(-%-GeometricCDFG6$%\"pG9$F(F(F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Below you
 will find three different ways to calculate P( " }{XPPEDIT 18 0 "X <=
3" "6#1%\"XG\"\"$" }{TEXT -1 6 " ) .  " }}{PARA 0 "" 0 "" {MPLTEXT 0 
21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(f(1)+f(2)+f
(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$v)!\"$" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(sum(f(x),x=1..3));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"++++]()!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "evalf(F(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++
++]()!#5" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Properties of the D
istribution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 267 61 "Calculation of Mean, Variance, and Moment Generating Fun
ction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Calculati
ng E(" }{TEXT 268 1 "X" }{TEXT -1 45 "), the expectation or mean of th
e Geometric( " }{TEXT 302 1 "p" }{TEXT -1 15 ") distribution." }}
{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 
"with(plots, display):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f
:=x->GeometricPDF(p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%
\"xG6\"6$%)operatorG%&arrowGF(-%-GeometricPDFG6$%\"pG9$F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "EX:=simplify(sum(x*f(x),x=1.
.infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EXG*&\"\"\"F&%\"pG!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 16 "Calculating Var(" }{TEXT 269 1 "X" }{TEXT -1 34 "), the varianc
e of the Geometric( " }{TEXT 303 1 "p" }{TEXT -1 15 ") distribution." 
}}{PARA 0 "" 0 "" {TEXT -1 33 "We will employ the formula:  Var(" }
{TEXT 270 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=1..infinity));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'E_X_SQG,$*&,&%\"pG\"\"\"!\"#F)
\"\"\"*$)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*\"\"#F+!\"\"F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The Momen
t Generating Function (MGF) can be easily calculated via Maple." }}
{PARA 0 "" 0 "" {TEXT -1 59 "Recall the moment generating function of \+
a random variable " }{TEXT 308 1 "X" }{TEXT -1 11 " is defined" }}
{PARA 0 "" 0 "" {TEXT -1 3 "as " }{TEXT 271 1 "M" }{TEXT -1 1 "(" }
{TEXT 272 3 " t " }{TEXT -1 6 ") = E(" }{XPPEDIT 18 0 "exp(tX)" "6#-%$
expG6#%#tXG" }{TEXT -1 36 "), provided this expectation 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=1..infinity));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&-%$expG6#%\"tG\"\"\"%\"pGF)F),(
\"\"\"F,F%!\"\"*&F%F,F*F,F,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "So the moment generating function \+
for a Geometric( " }{TEXT 304 1 "p" }{TEXT -1 18 ") random variable " 
}}{PARA 0 "" 0 "" {TEXT -1 11 "is given by" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "    " }{TEXT 273 1 "M" }{TEXT 
-1 2 "( " }{TEXT 274 1 "t" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "exp(t)*
p/(1-exp(t)+exp(t)*p);" "6#*(-%$expG6#%\"tG\"\"\"%\"pGF(,(\"\"\"F(-F%6
#F'!\"\"*&-F%6#F'F(F)F(F(F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "We will define " }{TEXT 275 1 "
M" }{TEXT -1 2 "( " }{TEXT 276 1 "t" }{TEXT -1 20 " ) as a function of
 " }{TEXT 277 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "M:=t->
exp(t)*p/(1-exp(t)+exp(t)*p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"M
GR6#%\"tG6\"6$%)operatorG%&arrowGF(*&*&-%$expG6#9$\"\"\"%\"pGF2F2,(\"
\"\"F5F.!\"\"*&F.F5F3F5F5!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The moment generating function \+
provides us alternative ways to calculate the" }}{PARA 0 "" 0 "" 
{TEXT -1 40 "mean and variance by way of the formula." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "M
^r;" "6#)%\"MG%\"rG" }{TEXT -1 9 " (0) = E(" }{XPPEDIT 18 0 "X^r" "6#)
%\"XG%\"rG" }{TEXT -1 10 ") , where " }{XPPEDIT 18 0 "M^r" "6#)%\"MG%
\"rG" }{TEXT -1 1 "(" }{TEXT 278 1 "t" }{TEXT -1 14 ") denotes the " }
{TEXT 279 1 "r" }{TEXT -1 18 " th derivative of " }{TEXT 280 1 "M" }
{TEXT -1 1 "(" }{TEXT 281 1 "t" }{TEXT -1 18 ") with respect to " }
{TEXT 282 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 30 "This formula holds as long as " }{TEXT 
283 1 "M" }{TEXT -1 1 "(" }{TEXT 284 1 "t" }{TEXT -1 45 ") exists in a
n open interval containing zero." }}{PARA 0 "" 0 "" {TEXT -1 79 "See, \+
for example, Mathematical Statistics and Data Analysis by John A. Rice
 for" }}{PARA 0 "" 0 "" {TEXT -1 39 "more on the moment generating fun
ction." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 18 "M_p:=diff(M(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$M_pG,&*&*&-%$expG6#%\"tG\"\"\"%\"pGF,F,,(\"\"\"F/F(!\"\"*&F(F
/F-F/F/!\"\"F/*&*(F(F,F-F,,&F(F0F1F/F/F,*$)F.\"\"#F,F2F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(M_p);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&*&-%$expG6#%\"tG\"\"\"%\"pGF)F)*$),(\"\"\"F.F%!\"\"*&
F%F.F*F.F.\"\"#F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "s
implify(subs(t=0,M_p));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$%
\"pG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 17 "And therefore if " }{TEXT 285 1 "X" }{TEXT -1 17 " is a G
eometric( " }{TEXT 305 1 "p" }{TEXT -1 19 ") variable, then E(" }
{TEXT 286 1 "X" }{TEXT -1 6 ") = 1/" }{TEXT 306 1 "p" }{TEXT -1 19 ", \+
which agrees with" }}{PARA 0 "" 0 "" {TEXT -1 57 "what we found earlie
r.  Now turning to the second moment." }}{PARA 0 "" 0 "" {MPLTEXT 0 
21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M_pp:=diff(M_p,t)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%M_ppG,(*&*&-%$expG6#%\"tG\"\"
\"%\"pGF,F,,(\"\"\"F/F(!\"\"*&F(F/F-F/F/!\"\"F/*&*(F(F,F-F,,&F(F0F1F/F
/F,*$)F.\"\"#F,F2!\"$*&*(F(F,F-F,)F5\"\"#F,F,*$)F.\"\"$F,F2F=" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(subs(t=0,M_pp));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"pG\"\"\"!\"#F'\"\"\"*$)F&\"
\"#F)!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Th
erefore E( " }{XPPEDIT 18 0 "X^2" "6#*$%\"XG\"\"#" }{TEXT -1 5 " ) = \+
" }{XPPEDIT 18 0 "(2-p)/(p^2);" "6#*&,&\"\"#\"\"\"%\"pG!\"\"F&*$F'\"\"
#F(" }{TEXT -1 45 " , which again is in agreement with the value" }}
{PARA 0 "" 0 "" {TEXT -1 64 "calculated previously. The variance is no
w quickly calculated as" }}{PARA 0 "" 0 "" {TEXT -1 4 "Var(" }{TEXT 
287 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 "(2
-p)/(p^2)-(1/p)^2;" "6#,&*&,&\"\"#\"\"\"%\"pG!\"\"F'*$F(\"\"#F)F'*$*&
\"\"\"F'F(F)\"\"#F)" }}{PARA 0 "" 0 "" {TEXT -1 14 "            = " }
{XPPEDIT 18 0 "(1-p)/p^2" "6#*&,&\"\"\"\"\"\"%\"pG!\"\"F&*$F'\"\"#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 -1 285 "We will now simu
late some data from a geometric distribution in order to compare the s
ampling distribution to the theoretical distribution.  The next bit of
 code will simulate a random sample of size 500 from a Geometric(0.5) \+
distribution, but you are invited to change the parameters." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r
estart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "p:=
0.5: numsim:=500:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sample
:=GeometricS(p,numsim):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "
a:=Histogram(sample,0.5..10+0.5,10):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "b:=ProbHist(GeometricPDF(p,x),1..10,10):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display([a,b],title=\"Simulated vs.
 Theoretical\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 
2 "6'-%'CURVESG6%7J7$$\"1+++++++]!#;\"\"!7$F($\"1+++++++^F*7$$\"1+++++
++:!#:F-7$F0F+F37$F0$\"1++++++?CF*7$$\"1+++++++DF2F57$F8F+F:7$F8$\"1++
++++?8F*7$$\"1+++++++NF2F<7$F?F+FA7$F?$\"1+++++++a!#<7$$\"1+++++++XF2F
C7$FGF+FI7$FG$\"1+++++++CFE7$$\"1+++++++bF2FK7$FNF+FP7$FN$\"1+++++++9F
E7$$\"1+++++++lF2FR7$FUF+FW7$FU$\"1+++++++7FE7$$\"1+++++++vF2FY7$FfnF+
Fhn7$Ffn$\"1+++++++!)!#=7$$\"1+++++++&)F2Fjn7$F^oF+F`o7$F^o$\"1+++++++
SF\\o7$$\"1+++++++&*F2Fbo7$FeoF+FgoFgo7$$\"1++++++]5!#9F+Fho-%'COLOURG
6&%$RGBG$\"*++++\"!\")F+F+-%*LINESTYLEG6#\"\"$-F$6$7JF'7$F(F(7$F0F(F3F
37$F0$F9F*7$F8F]qF:F:7$F8$\"1++++++]7F*7$F?F`qFAFA7$F?$\"1++++++]iFE7$
FGFdqFIFI7$FG$\"1++++++DJFE7$FNFhqFPFP7$FN$\"1+++++]i:FE7$FUF\\rFWFW7$
FU$\"1+++++]7yF\\o7$FfnF`rFhnFhn7$Ffn$\"1+++++D1RF\\o7$F^oFdrF`oF`o7$F
^o$\"1++++]7`>F\\o7$FeoFhrFgoFgo7$Feo$\"1++++]il(*!#>7$FioF\\sFho-F]p6
&F_pF+F+F`p-%+AXESLABELSG6$%!GFes-%&TITLEG6#Q:Simulated~vs.~Theoretica
l6\"-%%VIEWG6$%(DEFAULTGF^t" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Now let's compare \+
the simulated mean to the theoretical mean.  " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "theo_mean:=1/p;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%*theo_meanG$\"+++++?!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "sim_mean:=evalf(Mean(sample));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%)sim_meanG$\"++++%*>!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 47 "relative_error:=(sim_mean-theo_mean)/theo_mean;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%/relative_errorG$!+++++I!#7" }}}}}{MARK "3" 0 }
{VIEWOPTS 1 1 0 1 1 1803 }