{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 "" 1 18 0 
0 0 0 0 1 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 "" 1 18 0 0 0 0 0 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 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 
0 0 }{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 "" 0 1 0 0 0 0 1 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 }{CSTYLE "" -1 288 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 
0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 
-1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 
1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE 
"Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }
1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "
" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 
-1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 
0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{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 21 "with(plots, display):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "                    \+
  " }{TEXT 256 34 " The Discrete Uniform Distribution" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "The discrete uniform d
istribution on the set of integers \{1,2,3,...," }{TEXT 264 1 "N" }
{TEXT -1 22 "\} gives probability 1/" }{TEXT 265 1 "N" }{TEXT -1 3 " t
o" }}{PARA 0 "" 0 "" {TEXT -1 77 "each element of this set.  This dist
ribution is denoted by Discrete Uniform (" }{TEXT 283 1 "N" }{TEXT -1 
2 ")." }}{PARA 0 "" 0 "" {TEXT -1 65 "Hence, the probability distribut
ion function will be constant at " }{TEXT 266 2 "f " }{TEXT -1 1 "(" }
{TEXT 267 1 "x" }{TEXT -1 6 ") = 1/" }{TEXT 268 1 "N" }{TEXT -1 2 ", \+
" }{TEXT 288 1 "x" }{TEXT -1 15 " in \{1,2,3,...," }{TEXT 289 1 "N" }
{TEXT -1 2 "\}," }}{PARA 0 "" 0 "" {TEXT -1 52 "and the cumulative dis
tribution function is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 29 "              0,         if  " }{TEXT 
273 1 "x" }{TEXT -1 2 "<1" }}{PARA 0 "" 0 "" {TEXT 269 1 "F" }{TEXT 
-1 1 "(" }{TEXT 270 1 "x" }{TEXT -1 8 ") =    [" }{TEXT 271 1 "x" }
{TEXT -1 2 "]/" }{TEXT 272 1 "N" }{TEXT -1 12 ",   if  1<= " }{TEXT 
274 1 "x" }{TEXT -1 4 " <= " }{TEXT 275 1 "N" }}{PARA 0 "" 0 "" {TEXT 
-1 29 "              1,         if  " }{TEXT 276 1 "x" }{TEXT -1 1 ">
" }{TEXT 277 1 "N" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 7 "where [" }{TEXT 278 1 "x" }{TEXT -1 77 "] \+
denotes the greatest integer function (also known as the \"floor\" fun
ction)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 
"If " }{TEXT 279 1 "X" }{TEXT -1 62 " is the value showing on a roll o
f a fair six-sided die, then " }{TEXT 280 1 "X" }{TEXT -1 38 " has the
 discrete uniform distribution" }}{PARA 0 "" 0 "" {TEXT -1 103 "on the
 set \{1,2,3,4,5,6\}.  The following Maple code will form a list consi
sting of the distribution of " }{TEXT 281 1 "X" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 50 "DiscUnif_6:=[1,1/6,2,1/6,3,1/6,4,1/6,5,1/6,6,1/6];" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%+DiscUnif_6G7.\"\"\"#F&\"\"'\"\"#F'\"\"$F'\"\"
%F'\"\"&F'F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Th
e following Maple code will make a probability histogram for the distr
ibution of " }{TEXT 282 1 "X" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ProbHist(DiscUnif_6);" }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7:7$
$\"1+++++++]!#;\"\"!7$F($\"1nmmmmmm;F*7$$\"1+++++++:!#:F-7$F0F+F3F/7$$
\"1+++++++DF2F-7$F5F+F7F47$$\"1+++++++NF2F-7$F9F+F;F87$$\"1+++++++XF2F
-7$F=F+F?F<7$$\"1+++++++bF2F-7$FAF+FCF@7$$\"1+++++++lF2F-7$FEF+-%'COLO
URG6&%$RGBGF+F+$\"*++++\"!\")-%+AXESLABELSG6$%!GFR-%%VIEWG6$%(DEFAULTG
FV" 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 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 31 "Sample Probability Calculations" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "The following Maple code \+
will make a plot of the cumulative distribution function of " }{TEXT 
284 2 "X," }{TEXT -1 7 " where " }{TEXT 291 1 "X" }{TEXT -1 64 " is a \+
discrete uniform random variable on the set \{1,2,3,4,5,6\}." }}{PARA 
0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot(floor(x)/6,x=0.
.6.5,color=blue,discont=true);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
300 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$\"1+++++++?!#C\"\"!7$$\"10y%zg:(z@
!#<F+7$$\"1P#G&HVFwSF/F+7$$\"1b+$o'H:4iF/F+7$$\"1))>C3(phN)F/F+7$$\"1S
hQ0#)H\\5!#;F+7$$\"1Mq#H13uC\"F<F+7$$\"1U)RaMRDX\"F<F+7$$\"1@zD6kok;F<
F+7$$\"1)QX(=J:w=F<F+7$$\"1V'e*>En$4#F<F+7$$\"16c-:RE&G#F<F+7$$\"1*>'*
\\L]4]#F<F+7$$\"15*HTQAvr#F<F+7$$\"1\"3&H$oHi#HF<F+7$$\"1VOX0fv:JF<F+7
$$\"1])o\\\"47TLF<F+7$$\"1&>0#RY.KNF<F+7$$\"1\\NB'o7Tv$F<F+7$$\"1v0`(Q
*o]RF<F+7$$\"1SXe%=lj;%F<F+7$$\"1UIEY&R<P%F<F+7$$\"1H#*[Zh-'e%F<F+7$$
\"1,@XS\"3Gy%F<F+7$$\"15k=/T1&*\\F<F+7$$\"18X0(RQb@&F<F+7$$\"1_,7<>Y2a
F<F+7$$\"1*o2ndWZh&F<F+7$$\"1[WoYy))GeF<F+7$$\"1fkMe_QQgF<F+7$$\"1hcGw
Z3TiF<F+7$$\"13ajJ![hY'F<F+7$$\"1B)fmPx$omF<F+7$$\"1*ziuO+V)oF<F+7$$\"
1Jof*oe*zqF<F+7$$\"1TX#e\"\\'QH(F<F+7$$\"1(z-Y+M^\\(F<F+7$$\"1FzU&>=bq
(F<F+7$$\"1/&)o'>27\"zF<F+7$$\"1@Qu=XaE\")F<F+7$$\"134L`*RRL)F<F+7$$\"
1SD)*R<.Y&)F<F+7$$\"1lyb*Hnjv)F<F+7$$\"1D9?MQk\\*)F<F+7$$\"1#p[Tbg6<*F
<F+7$$\"1:&*=\\xGp$*F<F+7$$\"1qy#HmK0e*F<F+7$$\"1<\">'**4s#y*F<F+7$$\"
0+++)********!#:F+-%'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-F$6$7S7$$\"1+++
/+++5Fht$\"1nmmmmmm;F<7$$\"1!H#*z:(z@5FhtFfu7$$\"1JA8XFwS5FhtFfu7$$\"1
N*>9`\"4i5FhtFfu7$$\"1t\"[()phN3\"FhtFfu7$$\"1&*)=@#)H\\5\"FhtFfu7$$\"
1SPz23uC6FhtFfu7$$\"1pH'f$RDX6FhtFfu7$$\"1=*fCkok;\"FhtFfu7$$\"1yS78`h
(=\"FhtFfu7$$\"1&ReJEn$47FhtFfu7$$\"1c%)e#RE&G7FhtFfu7$$\"1S#*\\M]4]7F
htFfu7$$\"1#)fKRAvr7FhtFfu7$$\"1;!f\"pHi#H\"FhtFfu7$$\"1i!*H\"fv:J\"Fh
tFfu7$$\"1P/;#47TL\"FhtFfu7$$\"11x]kM?`8FhtFfu7$$\"15<7p7Tv8FhtFfu7$$
\"1#ys\"R*o]R\"FhtFfu7$$\"13>z=lj;9FhtFfu7$$\"13w([&R<P9FhtFfu7$$\"17X
\"\\h-'e9FhtFfu7$$\"1(3KT\"3Gy9FhtFfu7$$\"1:1U5k]*\\\"FhtFfu7$$\"1O#>'
RQb@:FhtFfu7$$\"1I!\\:>Y2a\"FhtFfu7$$\"1r[UdWZh:FhtFfu7$$\"1!*oL%y))Ge
\"FhtFfu7$$\"1#H>a_QQg\"FhtFfu7$$\"1K@8x%3Ti\"FhtFfu7$$\"1\"3xD![hY;Fh
tFfu7$$\"1J')*ptPom\"FhtFfu7$$\"1gD*f.I%)o\"FhtFfu7$$\"1+x7oe*zq\"FhtF
fu7$$\"13\\m!\\'QH<FhtFfu7$$\"1EAY*R8&\\<FhtFfu7$$\"1&eg%==bq<FhtFfu7$
$\"1nV]=27\"z\"FhtFfu7$$\"1kPi]Wl7=FhtFfu7$$\"1:&**R*RRL=FhtFfu7$$\"1T
)zD<.Y&=FhtFfu7$$\"1SKXGnjv=FhtFfu7$$\"1&GS=Qk\\*=FhtFfu7$$\"10ku`g6<>
FhtFfu7$$\"1O7<t(Gp$>FhtFfu7$$\"1u0YkK0e>FhtFfu7$$\"1B)[!)4s#y>FhtFfu7
$$\"1+++'*******>FhtFfuFit-F$6$7S7$$\"1+++5+++SFht$\"1mmmmmmmmF<7$$\"1
B2tjrz@SFhtF^_l7$$\"1yIk]FwSSFhtF^_l7$$\"1O[nO:4iSFhtF^_l7$$\"1Lau.<c$
3%FhtF^_l7$$\"1P(fo#)H\\5%FhtF^_l7$$\"1]oH73uCTFhtF^_l7$$\"1@*>-%RDXTF
htF^_l7$$\"1%Hikkok;%FhtF^_l7$$\"1%psoJ:w=%FhtF^_l7$$\"1))fkmsO4UFhtF^
_l7$$\"1Rh%eRE&GUFhtF^_l7$$\"1+\")\\P]4]UFhtF^_l7$$\"1b\\1UAvrUFhtF^_l
7$$\"1TvkrHi#H%FhtF^_l7$$\"1b,c$fv:J%FhtF^_l7$$\"1#4^T47TL%FhtF^_l7$$
\"1k#piY.KN%FhtF^_l7$$\"1vnhq7TvVFhtF^_l7$$\"1b>VS*o]R%FhtF^_l7$$\"1pA
z>lj;WFhtF^_l7$$\"1@:jbR<PWFhtF^_l7$$\"1\"G6ah-'eWFhtF^_l7$$\"1<FR93Gy
WFhtF^_l7$$\"1QlU5k]*\\%FhtF^_l7$$\"1!fg$RQb@XFhtF^_l7$$\"1x+1\">Y2a%F
htF^_l7$$\"1xrocWZhXFhtF^_l7$$\"1CAM$y))Ge%FhtF^_l7$$\"1IK<C&QQg%FhtF^
_l7$$\"1JGkv%3Ti%FhtF^_l7$$\"1/x\"3![hYYFhtF^_l7$$\"1yl*\\tPom%FhtF^_l
7$$\"1*RJP.I%)o%FhtF^_l7$$\"1\\<jle*zq%FhtF^_l7$$\"1qA\"z['QHZFhtF^_l7
$$\"1l!okR8&\\ZFhtF^_l7$$\"1jR@:=bqZFhtF^_l7$$\"1>4,:27\"z%FhtF^_l7$$
\"16>(oWaE\"[FhtF^_l7$$\"1(y)***)RRL[FhtF^_l7$$\"1.YKoJga[FhtF^_l7$$\"
1*fXRsOc([FhtF^_l7$$\"1725xV'\\*[FhtF^_l7$$\"175u[g6<\\FhtF^_l7$$\"1\"
4GzwGp$\\FhtF^_l7$$\"1MR'*eK0e\\FhtF^_l7$$\"1e&4B4s#y\\FhtF^_l7$$\"1++
+!*******\\FhtF^_lFit-F$6$7S7$$\"1+++7+++]Fht$\"1LLLLLLL$)F<7$$\"1NNkl
rz@]FhtFfhl7$$\"1G+[_FwS]FhtFfhl7$$\"1qkUQ:4i]FhtFfhl7$$\"1'=6aqhN3&Fh
tFfhl7$$\"1=+WG)H\\5&FhtFfhl7$$\"1))yz83uC^FhtFfhl7$$\"11*Q;%RDX^FhtFf
hl7$$\"1>kzZ'ok;&FhtFfhl7$$\"1LA7=`h(=&FhtFfhl7$$\"1>&3yEn$4_FhtFfhl7$
$\"1L?$pRE&G_FhtFfhl7$$\"1?x\\Q]4]_FhtFfhl7$$\"1Yz(HC_<F&FhtFfhl7$$\"1
\\qZsHi#H&FhtFfhl7$$\"1`QJ%fv:J&FhtFfhl7$$\"1VY\"[47TL&FhtFfhl7$$\"1]k
&oY.KN&FhtFfhl7$$\"1H^6r7Tv`FhtFfhl7$$\"1z;&3%*o]R&FhtFfhl7$$\"1Bd7?lj
;aFhtFfhl7$$\"1DG)e&R<PaFhtFfhl7$$\"1qod:EgeaFhtFfhl7$$\"1%fzW\"3GyaFh
tFfhl7$$\"18&G/T1&*\\&FhtFfhl7$$\"1uVFRQb@bFhtFfhl7$$\"1\"4(*3>Y2a&Fht
Ffhl7$$\"1!GTkXu9c&FhtFfhl7$$\"1p1,$y))Ge&FhtFfhl7$$\"1vyvB&QQg&FhtFfh
l7$$\"1(RY^Z3Ti&FhtFfhl7$$\"1W7B+[hYcFhtFfhl7$$\"1G#HVtPom&FhtFfhl7$$
\"1!oxH.I%)o&FhtFfhl7$$\"1m(*zke*zq&FhtFfhl7$$\"1CZ*p['QHdFhtFfhl7$$\"
17+Z&R8&\\dFhtFfhl7$$\"1c<89=bqdFhtFfhl7$$\"1Ok%Qr?6z&FhtFfhl7$$\"1$H@
cWaE\"eFhtFfhl7$$\"16_m))RRLeFhtFfhl7$$\"1!>1p;.Y&eFhtFfhl7$$\"1_IWAnj
veFhtFfhl7$$\"1b3_vV'\\*eFhtFfhl7$$\"1[D2Zg6<fFhtFfhl7$$\"1w.=m(Gp$fFh
tFfhl7$$\"1A<8dK0efFhtFfhl7$$\"1pkR!4s#yfFhtFfhl7$$\"1+++))******fFhtF
fhlFit-F$6$7S7$$\"1+++8+++gFht$\"\"\"F+7$$\"1r**z*e)*3,'FhtF^bm7$$\"1^
#*>$Q\"Q?gFhtF^bm7$$\"1W6:wd/JgFhtF^bm7$$\"1J?if3yTgFhtF^bm7$$\"1z]6@
\\Y_gFhtF^bm7$$\"1.Ux8/PigFhtF^bm7$$\"1*>uw(pisgFhtF^bm7$$\"1U<tIVB$3'
FhtF^bm7$$\"1,N(em2Q4'FhtF^bm7$$\"1#*[pSOo/hFhtF^bm7$$\"1!\\P_?jU6'Fht
F^bm7$$\"1l()*f_Z]7'FhtF^bm7$$\"1@s@Gh(e8'FhtF^bm7$$\"1-f%H\\6j9'FhtF^
bm7$$\"1]`%Q!yybhFhtF^bm7$$\"15K2ag0nhFhtF^bm7$$\"1@]2S<gwhFhtF^bm7$$
\"1a@=Ucq(='FhtF^bm7$$\"1r2.xW`(>'FhtF^bm7$$\"1Dikm#=$3iFhtF^bm7$$\"1Q
U]%)pe=iFhtF^bm7$$\"1L)HVJ,$HiFhtF^bm7$$\"1<:w8/9RiFhtF^bm7$$\"1]Zr6Kv
\\iFhtF^bm7$$\"1Mc6EpxgiFhtF^bm7$$\"1+y!>5t.F'FhtF^bm7$$\"1l\"fYBP2G'F
htF^bm7$$\"1XC#zRW9H'FhtF^bm7$$\"1*4v#o#>>I'FhtF^bm7$$\"1!4\\RCa?J'Fht
F^bm7$$\"13!pkS2LK'FhtF^bm7$$\"1wx\\t)=ML'FhtF^bm7$$\"15/!G-:UM'FhtF^b
m7$$\"1()=pQz*RN'FhtF^bm7$$\"1wzw\\KpkjFhtF^bm7$$\"1$\\&)RqcZP'FhtF^bm
7$$\"1F`H8fF&Q'FhtF^bm7$$\"1(4KJOgbR'FhtF^bm7$$\"1#*z**GsK1kFhtF^bm7$$
\"16#*\\+qp;kFhtF^bm7$$\"1#\\)f*e,tU'FhtF^bm7$$\"1*)eMn$=yV'FhtF^bm7$$
\"1ja'Q>#[ZkFhtF^bm7$$\"1e\">'H!e&ekFhtF^bm7$$\"1fK:*Qk%okFhtF^bm7$$\"
12ygMm-zkFhtF^bm7$$\"1j*>70O\"*['FhtF^bm7$$\"1+++++++lFhtF^bmFit-F$6$7
S7$$\"1+++1+++?Fht$\"1LLLLLLLLF<7$$\"1,^!*frz@?FhtFf[n7$$\"1\"=ppui2/#
FhtFf[n7$$\"1o:<L:4i?FhtFf[n7$$\"1ERT+<c$3#FhtFf[n7$$\"1v\"*pB)H\\5#Fh
tFf[n7$$\"1xZH43uC@FhtFf[n7$$\"1`>QPRDX@FhtFf[n7$$\"1VSzV'ok;#FhtFf[n7
$$\"1;OP9`h(=#FhtFf[n7$$\"1E4KksO4AFhtFf[n7$$\"1]Vn$RE&GAFhtFf[n7$$\"1
g))\\N]4]AFhtFf[n7$$\"1t*Q-C_<F#FhtFf[n7$$\"1C&))*pHi#H#FhtFf[n7$$\"1g
F0#fv:J#FhtFf[n7$$\"1))R#G47TL#FhtFf[n7$$\"1#*[4lM?`BFhtFf[n7$$\"1l+ip
7TvBFhtFf[n7$$\"11DfR*o]R#FhtFf[n7$$\"1i`7>lj;CFhtFf[n7$$\"17*G^&R<PCF
htFf[n7$$\"1-,3:EgeCFhtFf[n7$$\"1k*=U\"3GyCFhtFf[n7$$\"1!fA/T1&*\\#Fht
Ff[n7$$\"1@I`RQb@DFhtFf[n7$$\"1YgQ\">Y2a#FhtFf[n7$$\"1t*yrXu9c#FhtFf[n
7$$\"1N`+%y))Ge#FhtFf[n7$$\"1QR+D&QQg#FhtFf[n7$$\"1)pNmZ3Ti#FhtFf[n7$$
\"1A1*>![hYEFhtFf[n7$$\"1\"GJjtPom#FhtFf[n7$$\"1S)Q_.I%)o#FhtFf[n7$$\"
1;dHne*zq#FhtFf[n7$$\"1itu*['QHFFhtFf[n7$$\"1sTY)R8&\\FFhtFf[n7$$\"1y$
yt\"=bqFFhtFf[n7$$\"1&))Rtr?6z#FhtFf[n7$$\"1YJP\\Wl7GFhtFf[n7$$\"1Rfm#
*RRLGFhtFf[n7$$\"1H9;rJgaGFhtFf[n7$$\"1$p]psOc(GFhtFf[n7$$\"1F/E!Qk\\*
GFhtFf[n7$$\"1Tz2_g6<HFhtFf[n7$$\"1@NUr(Gp$HFhtFf[n7$$\"1h$GEE`!eHFhtF
f[n7$$\"1Nd8'4s#yHFhtFf[n7$$\"1+++%*******HFhtFf[nFit-F$6$7S7$$\"1+++3
+++IFht$\"1+++++++]F<7$$\"17z\"=;(z@IFhtF^en7$$\"1Ih!)[FwSIFhtF^en7$$
\"1-K#\\`\"4iIFhtF^en7$$\"1!oz?qhN3$FhtF^en7$$\"1d%z_#)H\\5$FhtF^en7$$
\"19ez53uCJFhtF^en7$$\"1P4!)QRDXJFhtF^en7$$\"1p\"G^kok;$FhtF^en7$$\"1b
Ji:`h(=$FhtF^en7$$\"1dM[lsO4KFhtF^en7$$\"1X-w%RE&GKFhtF^en7$$\"1![)\\O
]4]KFhtF^en7$$\"1k>:TAvrKFhtF^en7$$\"1L!=3(Hi#H$FhtF^en7$$\"1dk!Gfv:J$
FhtF^en7$$\"1Sv[$47TL$FhtF^en7$$\"1y?olM?`LFhtF^en7$$\"1?%=,F6aP$FhtF^
en7$$\"1IA,S*o]R$FhtF^en7$$\"1;)e%>lj;MFhtF^en7$$\"1<-QbR<PMFhtF^en7$$
\"1#pX_h-'eMFhtF^en7$$\"1SeI93GyMFhtF^en7$$\"1kXU5k]*\\$FhtF^en7$$\"10
oWRQb@NFhtF^en7$$\"1hIA\">Y2a$FhtF^en7$$\"1vI$pXu9c$FhtF^en7$$\"1zPn$y
))Ge$FhtF^en7$$\"1%e)eC&QQg$FhtF^en7$$\"1l#RhZ3Ti$FhtF^en7$$\"1jTS,[hY
OFhtF^en7$$\"1HRmNx$om$FhtF^en7$$\"1?^[M+V)o$FhtF^en7$$\"1LPYme*zq$Fht
F^en7$$\"1;)H))['QHPFhtF^en7$$\"1>hY(R8&\\PFhtF^en7$$\"1rhH;=bqPFhtF^e
n7$$\"1-a<;27\"z$FhtF^en7$$\"1HD7[Wl7QFhtF^en7$$\"1jBL\"*RRLQFhtF^en7$
$\"1;IupJgaQFhtF^en7$$\"1Y\"[asOc(QFhtF^en7$$\"1q0oyV'\\*QFhtF^en7$$\"
1w%4/0;r\"RFhtF^en7$$\"11enp(Gp$RFhtF^en7$$\"1[hzgK0eRFhtF^en7$$\"1YEA
%4s#yRFhtF^en7$$\"1+++#*******RFhtF^enFit-%+AXESLABELSG6$Q\"x6\"%!G-%%
VIEWG6$;F+$\"#l!\"\"%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We can ca
lculate probabilities for the Discrete Uniform (" }{TEXT 290 1 "N" }
{TEXT -1 43 ") distribution using either the probability" }}{PARA 0 "
" 0 "" {TEXT -1 90 "distribution function (PDF) or the cumulative dist
ribution function (CDF).  We will denote" }}{PARA 0 "" 0 "" {TEXT -1 
11 "the PDF by " }{TEXT 257 1 "f" }{TEXT -1 16 " and the CDF by " }
{TEXT 258 1 "F" }{TEXT -1 16 ".  The value of " }{TEXT 285 1 "N" }
{TEXT -1 61 " is set at 20, but you are encouraged to try other values
 of " }{TEXT 292 1 "N" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {MPLTEXT 0 
21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=20;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "f:=x->1/N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR
6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-%\"NG!\"\"F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "F:=x->floor(x)/N;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%
&floorG6#9$\"\"\"%\"NG!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Below you will find three differen
t ways to calculate P(4<= " }{TEXT 263 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#$\"+++++D!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "evalf(sum(f(x),x=4..8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++
++D!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(F(8)-F(3));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++D!#5" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Pro
perties of the Distribution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 259 32 "Calculation of Mean and Variance" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Calculating E(" }
{TEXT 260 1 "X" }{TEXT -1 52 "), the expectation or mean of the Discre
te Uniform (" }{TEXT 286 1 "N" }{TEXT -1 15 ") distribution." }}{PARA 
0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "with(plots, \+
display):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->1/N;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-%\"NG
!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "F:=x->floor(
x)/N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6#%\"xG6\"6$%)operator
G%&arrowGF(*&-%&floorG6#9$\"\"\"%\"NG!\"\"F(F(F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 33 "EX:=simplify(sum(x*f(x),x=1..N));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#EXG,&%\"NG#\"\"\"\"\"#F'F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Calculati
ng Var(" }{TEXT 261 1 "X" }{TEXT -1 41 "), the variance of the Discret
e Uniform (" }{TEXT 287 1 "N" }{TEXT -1 15 ") distribution." }}{PARA 
0 "" 0 "" {TEXT -1 33 "We will employ the formula:  Var(" }{TEXT 262 
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 39 "E_X_SQ:=simplify(sum(x^2*f(x),x=1..N));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%'E_X_SQG,(*$)%\"NG\"\"#\"\"\"#\"\"\"\"\"$F(#F,
F)#F,\"\"'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "VarX:=simpl
ify(E_X_SQ-EX^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VarXG,&*$)%\"N
G\"\"#\"\"\"#\"\"\"\"#7#!\"\"F-F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Simulation
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
321 "We will now simulate some data from an discrete uniform 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 Discrete Uniform distribution on the set \{1,2,3,...,20
\}, but you are invited to change the parameters." }}{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 19 "n:=20: numsim
:=500:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sample:=DiscUnifo
rmS(1..20,numsim):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "a:=Hi
stogram(sample,0.5..n+0.5,n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "b:=plot(1/n,0..n+1,color=blue):" }}}{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%7
\\p7$$\"1+++++++]!#;\"\"!7$F($\"1+++++++Y!#<7$$\"1+++++++:!#:F-7$F1F+F
47$F1$\"1+++++++kF/7$$\"1+++++++DF3F67$F9F+F;7$F9$\"1+++++++aF/7$$\"1+
++++++NF3F=7$F@F+FB7$F@$\"1+++++++[F/7$$\"1+++++++XF3FD7$FGF+FI7$FG$\"
1+++++++WF/7$$\"1+++++++bF3FK7$FNF+FP7$FNFD7$$\"1+++++++lF3FD7$FSF+FU7
$FS$\"1+++++++uF/7$$\"1+++++++vF3FW7$FZF+Ffn7$FZ$\"1+++++++QF/7$$\"1++
+++++&)F3Fhn7$F[oF+F]o7$F[oFD7$$\"1+++++++&*F3FD7$F`oF+Fbo7$F`oF-7$$\"
1++++++]5!#9F-7$FeoF+FhoFdo7$$\"1++++++]6FgoF-7$FjoF+F\\pFio7$$\"1++++
++]7FgoF-7$F^pF+F`p7$F^p$\"1+++++++gF/7$$\"1++++++]8FgoFbp7$FepF+Fgp7$
Fep$\"1+++++++eF/7$$\"1++++++]9FgoFip7$F\\qF+F^q7$F\\q$\"1+++++++KF/7$
$\"1++++++]:FgoF`q7$FcqF+Feq7$FcqFD7$$\"1++++++];FgoFD7$FhqF+Fjq7$Fhq$
\"1+++++++cF/7$$\"1++++++]<FgoF\\r7$F_rF+Far7$F_r$F)F/7$$\"1++++++]=Fg
oFcr7$FerF+Fgr7$FerFK7$$\"1++++++]>FgoFK7$FjrF+F\\s7$FjrFcr7$$\"1+++++
+]?FgoFcr7$F_sF+-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%*LINESTYLEG6#\"
\"$-F$6$7S7$F+Fcr7$$\"1+++vBSxXF*Fcr7$$\"1,+D1d<g&)F*Fcr7$$\"1++D'3ARI
\"F3Fcr7$$\"1++v.cza<F3Fcr7$$\"1+]7)>EN?#F3Fcr7$$\"1+]i+pb>EF3Fcr7$$\"
1+]i&fK.0$F3Fcr7$$\"1+]iN9%e\\$F3Fcr7$$\"1+]7B:#*RRF3Fcr7$$\"1++]xCr'R
%F3Fcr7$$\"1++v)>a!*z%F3Fcr7$$\"1++]#o&*>D&F3Fcr7$$\"1++]()pz1dF3Fcr7$
$\"1++]<B3XhF3Fcr7$$\"1+]i&Q(3VlF3Fcr7$$\"1++]2RN;qF3Fcr7$$\"1,++IFF<u
F3Fcr7$$\"1+]iImj$)yF3Fcr7$$\"1*****\\qZkH)F3Fcr7$$\"1,]i!)oO\\()F3Fcr
7$$\"1,](=/`1=*F3Fcr7$$\"1,+D1\\lI'*F3Fcr7$$\"1+DJ4(*Q/5FgoFcr7$$\"1+]
(=Yj*[5FgoFcr7$$\"1+DcjIE&4\"FgoFcr7$$\"1+v$H+nb8\"FgoFcr7$$\"1+]ihj4z
6FgoFcr7$$\"1++][k1C7FgoFcr7$$\"1+]7041o7FgoFcr7$$\"1+D1/yi58FgoFcr7$$
\"1+](y3\"*yN\"FgoFcr7$$\"1++]]#f.S\"FgoFcr7$$\"1++vyIqX9FgoFcr7$$\"1+
DcE8z'[\"FgoFcr7$$\"1++DM;rJ:FgoFcr7$$\"1+D1V\"yRd\"FgoFcr7$$\"1+DJB)e
\"=;FgoFcr7$$\"1++v8NNh;FgoFcr7$$\"1+Dc^Vd1<FgoFcr7$$\"1+++$RF,v\"FgoF
cr7$$\"1+]Pomm%z\"FgoFcr7$$\"1+D1Or$)Q=FgoFcr7$$\"1++]3_Uz=FgoFcr7$$\"
1+]()>P%f#>FgoFcr7$$\"1+++J/bn>FgoFcr7$$\"1+D1j=\">,#FgoFcr7$$\"1+v$RT
rV0#FgoFcr7$$\"#@F+Fcr-Fcs6&FesF+F+Ffs-%&TITLEG6#Q:Simulated~vs.~Theor
etical6\"-%+AXESLABELSG6$%!GF[^l-%%VIEWG6$;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 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Now let's compare the sim
ulated mean to the theoretical mean.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "theo_mean:=n/2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*theo_meanG\"#5" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "sim_mean:=evalf(Mean(sample));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%)sim_meanG$\"+++gL5!\")" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 47 "relative_error:=(sim_mean-theo_mean)/theo_mean
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/relative_errorG$\"++++gL!#6" }
}}}}{MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 }