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}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 43 "The Standard Normal or
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"" {TEXT -1 336 "The standard normal distribution is a single-peaked, \+
bell-shaped distribution that is centered at zero.  Many textbooks den
ote this distribution with N(0, 1) notation.  The zero indicates the c
enter of the distribution and the one indicates the standard deviation
.  The Maple code for sketching the standard normal curve is shown bel
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{TEXT 257 28 "General Normal Distributions" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 468 "There are many different distr
ibutions with the same overall shape as the N(0, 1) distribution.  The
se distributions may be centered at different locations or have a meas
ure of spread different from one, but they exhibit the bell-shaped pat
tern shown above.  Two parameters, one to identify the mean (mu) and a
nother to identify the variance (sigma^2), are needed to specify the p
robability distribution function for a general normal distribution.  M
ore specifically, " }}{PARA 258 "" 0 "" {TEXT -1 7 "f(x) = " }
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"" {TEXT -1 117 "A notation used to identify an arbitrary normal distr
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "A normal \+
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{XPPEDIT 18 0 "sigma^2" "6#*$%&sigmaG\"\"#" }{TEXT -1 32 " constant at
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" {TEXT -1 63 "The following animation will draw a sequence of normal \+
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increases from 1 to 31 in increments of 1 while holding " }{XPPEDIT 
18 0 "mu" "6#%#muG" }{TEXT -1 50 " constant at 0.  Do you see why stat
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*RuQ$osFju7$Fhy$\"1c.H3dZzsFju7$F]z$\"1[J$)=fi$G(Fju7$Fbz$\"1o@`Lk_!G(
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$F[\\l$\"1TQVN2.krFju7$Fe\\l$\"1o;0=0<!3(Fju7$Fi]l$\"1!>)RLErGoFju7$Fc
^l$\"1([,68lxZ'Fju7$Fg_l$\"1d&)=j8TfgFju7$Fa`l$\"1FV5tP%**f&Fju7$Ff`l$
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p#4\"Fju7$Fjcl$\"1\")pXaF8=&)Fet7$F_dl$\"1!=!=hW!Rb'Fet7$Fddl$\"1_+\"=
2ZG3&Fet7$Fidl$\"1PQ;'[d\"RPFet7$F_el$\"1=&H,(*y7!GFet7$Feel$\"1aF*\\6
O(G?Fet7$Fjel$\"12bl#*3ho9Fet7$F_flF\\`clFaflFeflF[gl7&-F(6%F*%.sigma2
~is~31.GF2-F:6$7Y7$F>$\"1B^Al&)[`6Fet7$FD$\"15RqCOTS;Fet7$FK$\"1K+%)HE
H+AFet7$FQ$\"1$GhAY)4=IFet7$FW$\"1I?R.E1'3%Fet7$Fgn$\"1QWG4T3TaFet7$F]
o$\"1ks2'*R%3+(Fet7$Fco$\"1L$oj(zBl*)Fet7$Fio$\"1k>8!>I59\"Fju7$F_p$\"
1cJizhyH9Fju7$Ffp$\"1E..H]kv<Fju7$F\\q$\"1+s,/(487#Fju7$Fbq$\"1Bg%=GlU
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1S3y))4#Q(fFju7$Ffu$\"1yN#RyX#)Q'Fju7$Fav$\"1pVJYt+8nFju7$Few$\"1(4LG)
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1RbS%H29(pFju7$Fi]l$\"1X+j&>g;t'Fju7$Fc^l$\"1r<_Izd'R'Fju7$Fg_l$\"1xq5
7wO'*fFju7$Fa`l$\"1-*yX,&zbbFju7$Ff`l$\"1&)Q_UB\"R-&Fju7$F[al$\"1%e!*z
5LX_%Fju7$F`al$\"1U9sw,7')RFju7$Feal$\"1[7/T/$o]$Fju7$F[bl$\"1c:49ft/I
Fju7$Fabl$\"1^W5E%GDc#Fju7$Ffbl$\"1%)yA&=())Q@Fju7$F[cl$\"1Vs\">3Rtw\"
Fju7$F`cl$\"1vyr.+\"eU\"Fju7$Fecl$\"1_.LZzvU6Fju7$Fjcl$\"1[Bg<!f-)*)Fe
t7$F_dl$\"1#o&Q6:9opFet7$Fddl$\"1$ynU\\+'[aFet7$Fidl$\"1)[Rf<9\"[SFet7
$F_el$\"15'em5D61$Fet7$Feel$\"1A()H:m6SAFet7$Fjel$\"1%H8=!*>'Q;Fet7$F_
flF\\[dlFaflFeflF[gl-%&TITLEG6#Q7Increasing~VariabilityFifl" 1 2 0 1 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{PARA 13 "" 1 "" {TEXT 
-1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Sample Probability Cal
culations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 155 "The cumulative distribution function (CDF) is used to co
mpute probabilities for normal distributions.  The CDF for the N(0,1) \+
distribution is shown below. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(NormalCDF(0,1,x),x=-4..
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F*-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fiz%(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 38 "a:=plot(f1, z=-3.5..3.5, color=black):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 51 "b:=plot(f1, z=-3.5..-1, color=yellow, filled=t
rue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "display([a, b]);" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
109 "Find the value of the yellow shaded area, the area under the stan
dard normal density curve to the left of -1." }}{PARA 13 "" 1 "" 
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Fj5F[p7$Fa]mFe]l7&Fe]mF`]m7$$!1+vV[/it:F*$\"1zl'3cCm:\"F[p7$Fh]mFe]l7&
F\\^mFg]m7$$!1nm\"z>)>A:F*$\"1:lVl:X_7F[p7$F_^mFe]l7&Fc^mF^^m7$$!1+v=n
QOo9F*$\"1.r5z'RuN\"F[p7$Ff^mFe]l7&Fj^mFe^m7$$!1LLL3]^;9F*$\"1oaL&y`GY
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(RQB3D>F[p7$Fi`mFe]l7&F]amFh`m7$$!1LLLe!yw:\"F*$\"1a5gp))>T?F[p7$F`amF
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$!1+DJX(>V0\"F*$\"1;:fFoR)G#F[p7$F^bmFe]l7&FbbmF]bm7$$!\"\"Fe]l$\"1M9>
Xsq>CF[p7$FebmFe]l7\"-%&STYLEG6#%,PATCHNOGRIDG-Fb]l6&Fd]l$\"*++++\"!\"
)FacmFe]l-%+AXESLABELSG6$Q\"z6\"%!G-%%VIEWG6$;$!#NFfbm$\"#NFfbm%(DEFAU
LTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "NormalCDF(0, 1, -1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+SDb'e\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Areas under normal curves can a
lso be found by using the integration command." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalf(int(N
ormalPDF(0,1,z), z=-infinity..-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#$\"+RDb'e\"!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "a:=plot(
f1, z=-3.5..3.5, color=black):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "b
:=plot(f1, z=.6..1.4, color=yellow, filled=true):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 16 "display([a, b]);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Find the value of the ye
llow shaded area, the area under the standard normal density curve bet
ween .6 and 1.4." }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 
2 "6&-%'CURVESG6$7en7$$!1+++++++N!#:$\"1/wX]p#os)!#>7$$!1ML$3#*>uM$F*$
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M7$$!1L$e9_>ZL#F*$\"1tbh*y=Ph#FM7$$!1+]iDGp'=#F*$\"1E%*)>oDDl$FM7$$!1m
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\"1Z\">Sj$*4R\"F[p7$$!1L$e9(3(*=8F*$\"1H1\"*G)Q;n\"F[p7$$!1mm;k`@h6F*$
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1t[pK/SFFF[p7$$!1nmm;V<XtF[p$\"1C$4l'3=YIF[p7$$!1'**\\7tVa$eF[p$\"1[EZ
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RF[p7$$\"1nTNr(QH7\"F[p$\"1PSX\"z[V'RF[p7$$\"1omT&yo(3:F[p$\"17H5hHFWR
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=F*$\"1(f$o\"\\<%QmFM7$$\"1LL$e/Xy.#F*$\"1#Ql,+Q>+&FM7$$\"1+](=<\"e)=#
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1)Qg-:,A$=FM7$$\"1M$3-7d%HEF*$\"1>\"y@?0wD\"FM7$$\"1,++&p]Zw#F*$\"1(*e
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Fg^l7$$\"1MLLBAt'\\'F[p$\"1%ok!3&4/B$F[p7$F__lFe]l7&Fc_lF^_l7$$\"1LLLj
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Fe]l7&Fh`lFc`l7$$\"1*****\\YJ?;(F[p$\"1dZ'Rq<p3$F[p7$F[alFe]l7&F_alFj`
l7$$\"1LLL=\"\\<L(F[p$\"1)G&*>X$=\\IF[p7$FbalFe]l7&FfalFaal7$$\"1+++&[
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\"1AX(*=Fm?CF[p7$FbhlFe]l7&FfhlFahl7$$\"1LL$=2Vs,\"F*$\"1RMVw\"))zP#F[
p7$FihlFe]l7&F]ilFhhl7$$\"1++]`pfK5F*$\"1Cl7l)f3M#F[p7$F`ilFe]l7&FdilF
_il7$$\"1LLLm&z\"\\5F*$\"1QEs<=!3I#F[p7$FgilFe]l7&F[jlFfil7$$\"1+++G5J
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1mmm!>qM8\"F*$\"1i3W8uf)4#F[p7$Fj[mFe]l7&F^\\mFi[m7$$\"1+++ISu]6F*$\"1
hz`43gd?F[p7$Fa\\mFe]l7&Fe\\mF`\\m7$$\"1LL$ep'Rm6F*$\"1'3M#4Ji??F[p7$F
h\\mFe]l7&F\\]mFg\\m7$$\"1+++%>4N=\"F*$\"1^sQ/9S!)>F[p7$F_]mFe]l7&Fc]m
F^]m7$$\"1nm;@2h*>\"F*$\"1NgyCxwU>F[p7$Ff]mFe]l7&Fj]mFe]m7$$\"1++]c9W;
7F*$\"1***ojGmO!>F[p7$F]^mFe]l7&Fa^mF\\^m7$$\"1nmmwl*GB\"F*$\"1#Rv2U(o
l=F[p7$Fd^mFe]l7&Fh^mFc^m7$$\"1++]iN7]7F*$\"11T^a(3i#=F[p7$F[_mFe]l7&F
__mFj^m7$$\"1LLL(>:nE\"F*$\"1D[0=UZ)y\"F[p7$Fb_mFe]l7&Ff_mFa_m7$$\"1LL
LSDo$G\"F*$\"1Ld30J>]<F[p7$Fi_mFe]l7&F]`mFh_m7$$\"1nm;&Q40I\"F*$\"1fQ!
*p7b7<F[p7$F``mFe]l7&Fd`mF_`m7$$\"1+++3:(fJ\"F*$\"1&*R].vDy;F[p7$Fg`mF
e]l7&F[amFf`m7$$\"1nmmXGpL8F*$\"1<%*GZYJR;F[p7$F^amFe]l7&FbamF]am7$$\"
1LLL@Ia\\8F*$\"1*Gl)\\G#[g\"F[p7$FeamFe]l7&FiamFdam7$$\"1++]9EWm8F*$\"
1HB4EBTo:F[p7$F\\bmFe]l7&F`bmF[bm7$$\"1++]\"o<EQ\"F*$\"1<C;;k#R`\"F[p7
$FcbmFe]l7&FgbmFbbm7$$\"1+++++++9F*$\"1\\uNcYF(\\\"F[p7$FjbmFe]l7\"-%&
STYLEG6#%,PATCHNOGRIDG-Fb]l6&Fd]l$\"*++++\"!\")FfcmFe]l-%+AXESLABELSG6
$Q\"z6\"%!G-%%VIEWG6$;$!#N!\"\"$\"#NFedm%(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 96 "Using the CDF, we find the area to
 the left of 1.4 and then subtract the area to the left of .6." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "area1:=NormalCDF(0,1,1.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
&area1G$\"+2MV#>*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "are
a2:=NormalCDF(0,1,.6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&area2G$\"
+A)ouD(!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "area_between:
=area1-area2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-area_betweenG$\"+&
ek\\$>!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 79 "Using the integration comand we simply integrate the N(0,
1) PDF from .6 to 1.4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "evalf(int(NormalPDF(0,1,z), z=.6..1
.4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'ek\\$>!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "a:=plot(f1, z=-3.5..3.5, color=blac
k, axes=NONE):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "b:=plot(f1, z=.67
45..3.5, color=yellow, filled=true, axes=NONE):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "c:=plot(x*0, x=-3.5..3.5, color=black, axes=NONE):" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "d:=textplot( [.68, -.02, \"z\"]):
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "e:=textplot( [1.04, .07, \".25
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" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 92 "Find the value of z.  z is known as the third quartile fo
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300 300 {PLOTDATA 2 "6*-%'CURVESG6$7en7$$!1+++++++N!#:$\"1/wX]p#os)!#>
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R)RC)e-#F*$\"1bI*3l1]7&FM7$F[hlFe]l7&F_hlFjgl7$$\"1#HKEObe3#F*$\"1PmK0
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FijlFdjl7$$\"1(=U\\]=zV#F*$\"1)4]I(==V?FM7$F\\[mFe]l7&F`[mF[[m7$$\"1Dc
*z75:]#F*$\"1$3)yRIAY<FM7$Fc[mFe]l7&Fg[mFb[m7$$\"1L3mH+leDF*$\"1`8K[P<
6:FM7$Fj[mFe]l7&F^\\mFi[m7$$\"1]i&42f'>EF*$\"1T3#*=YO!H\"FM7$Fa\\mFe]l
7&Fe\\mF`\\m7$$\"1aj4IB%\\n#F*$\"1e7tVno96FM7$Fh\\mFe]l7&F\\]mFg\\m7$$
\"1](eX`\"QNFF*$\"1@(f#))ocl%*F37$F_]mFe]l7&Fc]mF^]m7$$\"1@0#e6]Az#F*$
\"1*G8=b[))3)F37$Ff]mFe]l7&Fj]mFe]m7$$\"1Pf<pTp^GF*$\"1(pLb^_'RoF37$F]
^mFe]l7&Fa^mF\\^m7$$\"1$e9nf6)4HF*$\"1W3X3(p_y&F37$Fd^mFe]l7&Fh^mFc^m7
$$\"1(o/t:b1(HF*$\"1t,w(f4w$[F37$F[_mFe]l7&F__mFj^m7$$\"1n;ePZDHIF*$\"
1siZ*oLx0%F37$Fb_mFe]l7&Ff_mFa_m7$$\"1#zR@F\"=*3$F*$\"1B)3Y4U!yLF37$Fi
_mFe]l7&F]`mFh_m7$$\"1@bt%e6'[JF*$\"1M.H%=.l!GF37$F``mFe]l7&Fd`mF_`m7$
$\"1+v\")p=A.KF*$\"1mLa>BifBF37$Fg`mFe]l7&F[amFf`m7$$\"1e*)G>9\"eE$F*$
\"1$*zU1R<F>F37$F^amFe]l7&FbamF]am7$$\"1n;ZeAz@LF*$\"1(*4i[,m-;F37$Fea
mFe]l7&FiamFdam7$$\"1)=(34&z9Q$F*$\"1_zEN;378F37$F\\bmFe]l7&F`bmF[bm7$
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STYLEG6#%,PATCHNOGRIDG-Fb]l6&Fd]l$\"*++++\"!\")FacmFe]l-F$6$7S7$F(Fe]l
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]l7$FhnFe]l7$F]oFe]l7$FboFe]l7$FgoFe]l7$F]pFe]l7$FbpFe]l7$FgpFe]l7$F\\
qFe]l7$FaqFe]l7$FfqFe]l7$F[rFe]l7$F`rFe]l7$FerFe]l7$F_sFe]l7$FctFe]l7$
FguFe]l7$FavFe]l7$FfvFe]l7$F[wFe]l7$F`wFe]l7$FewFe]l7$FjwFe]l7$F_xFe]l
7$FdxFe]l7$FixFe]l7$F^yFe]l7$FcyFe]l7$FhyFe]l7$F]zFe]l7$FbzFe]l7$FgzFe
]l7$F\\[lFe]l7$Fa[lFe]l7$Ff[lFe]l7$F[\\lFe]l7$F`\\lFe]l7$Fe\\lFe]l7$Fj
\\lFe]lFibmFa]l-%%TEXTG6$7$$\"#o!\"#$F]gmF]gm%\"zG-Fhfm6$7$$\"$/\"F]gm
$\"\"(F]gm%$.25G-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q\"z6\"%!G-%%VIE
WG6$;$!#N!\"\"$\"#NFhhm%(DEFAULTG" 1 2 0 1 0 2 9 1 1 2 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 96 "Now, we know the area under the N(0,1) cu
rve to the left of z is .75 and we want to solve for z." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "fsolve
(NormalCDF(0,1,z)=.75,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/v*[u
'!#5" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Properties of the Distr
ibution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 61 "Calcula
tion of Mean, Variance, and Moment Generating Function" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "To calculate the mea
n of the Normal distribution,  we can simply integrate " }{TEXT 269 1 
"x" }{TEXT -1 62 "*(NormalPDF),  per the definition of mathematical ex
pectation." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "with(plots, display):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "interface(showassumed=0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "assume(sigma > 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "f(x):=NormalPDF(mu,sigma^2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>-%\"fG6#%\"xG,$*&*&-%%sqrtG6#\"\"#\"\"\"-%$expG6#,$*&*$),&F'\"\"\"%#m
uG!\"\"F.F/F/*$)%'sigma|irG\"\"#F/!\"\"#F:F.F8F/*&F=\"\"\"-F,6#%#PiGF/
F?#F8F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "EX:=int(x*f(x),x
=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EXG%#muG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
16 "Calculating Var(" }{TEXT 258 1 "X" }{TEXT -1 30 "), the variance o
f the Normal(" }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "sigma^2" "6#*$%&sigmaG\"\"#" }{TEXT -1 50 ") distributi
on.  We will employ the formula:  Var(" }{TEXT 259 1 "X" }{TEXT -1 7 "
) = E( " }{XPPEDIT 18 0 "X^2;" "6#*$%\"XG\"\"#" }{TEXT -1 5 " ) - " }
{XPPEDIT 18 0 "( E(X) )^2" "6#*$-%\"EG6#%\"XG\"\"#" }}{PARA 0 "" 0 "" 
{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "E_X_SQ
:=int((x^2)*f(x),x=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'E_X_SQG,&*$)%'sigma|irG\"\"#\"\"\"\"\"\"*$)%#muGF)F*F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "VarX:=simplify(E_X_SQ-EX^2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VarXG*$)%'sigma|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 generating function of a random variable " 
}{TEXT 267 1 "X" }{TEXT -1 15 " is defined as " }{TEXT 260 1 "M" }
{TEXT -1 1 "(" }{TEXT 261 3 " t " }{TEXT -1 6 ") = E(" }{XPPEDIT 18 0 
"exp(tX)" "6#-%$expG6#%#tXG" }{TEXT -1 36 "), provided this expectatio
n exists." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int(exp(t*x)*f(x),x=-infinit
y..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$*&%\"tG\"
\"\",&*&F(\"\"\")%'sigma|irG\"\"#F)F,%#muGF/F,#F,F/" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "So the moment g
enerating function for a Normal(" }{XPPEDIT 18 0 "mu" "6#%#muG" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "sigma^2" "6#*$%&sigmaG\"\"#" }{TEXT 
-1 29 ") random variable is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 4 "    " }{TEXT 262 1 "M" }{TEXT -1 2 "( " 
}{TEXT 263 1 "t" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "exp(mu*t+sigma^2*
t^2/2);" "6#-%$expG6#,&*&%#muG\"\"\"%\"tGF)F)*(%&sigmaG\"\"#F*\"\"#\"
\"#!\"\"F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 15 "We will define " }{TEXT 264 1 "M" }{TEXT -1 2 "( " }{TEXT 265 
1 "t" }{TEXT -1 20 " ) as a function of " }{TEXT 266 1 "t" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "M(t):=exp(mu*t+(sigma^2*t^2)/2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"MG6#%\"tG-%$expG6#,&*&%#muG\"\"
\"F'F.F.*&)%&sigmaG\"\"#\"\"\")F'F2F3#F.F2" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 122 "The moment generating function provides us with alternative wa
ys 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 270 1 "t" }{TEXT -1 14 ") denotes t
he " }{TEXT 271 1 "r" }{TEXT -1 18 " th derivative of " }{TEXT 272 1 "
M" }{TEXT -1 1 "(" }{TEXT 273 1 "t" }{TEXT -1 18 ") with respect to " 
}{TEXT 274 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 30 "This formula holds as long as " }{TEXT 
275 1 "M" }{TEXT -1 1 "(" }{TEXT 276 1 "t" }{TEXT -1 208 ") exists in \+
an open interval containing zero. See, for example, Mathematical Stati
stics and Data Analysis by John A. Rice for more on the moment generat
ing function.  Looking at the first derviative, we 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#>%$M_pG*&,&%#mu
G\"\"\"*&)%&sigmaG\"\"#\"\"\"%\"tGF(F(F(-%$expG6#,&*&F'F(F.F-F(*&F*F-)
F.F,F-#F(F,F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(M
_p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%#muG\"\"\"*&)%&sigmaG\"\"
#\"\"\"%\"tGF&F&F&-%$expG6#,$*&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#%#muG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 38 "This agrees with our previous result
, " }{XPPEDIT 18 0 "E(X)=mu" "6#/-%\"EG6#%\"XG%#muG" }{TEXT -1 45 ".  \+
Turning to the second derivative, we find:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M_pp:=diff(M_p,t)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%M_ppG,&*&)%&sigmaG\"\"#\"\"\"-
%$expG6#,&*&%#muG\"\"\"%\"tGF1F1*&F'F*)F2F)F*#F1F)F1F1*&),&F0F1*&F'F*F
2F*F1F)F*F+F*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify
(subs(t=0,M_pp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%&sigmaG\"\"
#\"\"\"\"\"\"*$)%#muGF'F(F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 12 "Therefore E(" }{XPPEDIT 18 0 "X^2" "6#*$
%\"XG\"\"#" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "sigma^2+mu^2" "6#,&*$%&
sigmaG\"\"#\"\"\"*$%#muG\"\"#F'" }{TEXT -1 44 ", which again is in agr
eement with the value" }}{PARA 0 "" 0 "" {TEXT -1 64 "calculated previ
ously. The variance is now quickly calculated as" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "Var(" }{TEXT 277 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 13 "          \+
 = " }{XPPEDIT 18 0 "sigma^2+mu^2-mu^2;" "6#,(*$%&sigmaG\"\"#\"\"\"*$%
#muG\"\"#F'*$F)\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 13 "           =
 " }{XPPEDIT 18 0 "sigma^2;" "6#*$%&sigmaG\"\"#" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Simulation" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 280 "W
e will now simulate some data from a normal distribution in order to c
ompare the sampling distribution to the theoretical distribution.  The
 next bit of code will simulate a random sample of size 500 from a Nor
mal(0, 1) distribution, but you are invited 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 24 "mu:=0: sigma:=1: n:=500:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 31 "sample:=NormalS(mu,sigma^2, n):" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "Remember that wh
en 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:=32:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a:=Histogram(sample,-4..4,num_bins)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "b:=plot(NormalPDF(mu,s
igma^2,x),x=-4..4,color=black):" }}}{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\\s7$$!\"
%\"\"!F*F'7$$!1++++++]P!#:F*F+F+F+7$$!1+++++++NF.F*F/F/F/7$$!1++++++]K
F.F*F2F27$F3$\"1+++++++!)!#=7$$!\"$F*F67$F:F*F<F97$$!1++++++]FF.F67$F>
F*F@F=7$$!1+++++++DF.F67$FBF*FD7$FB$\"1+++++++C!#<7$$!1++++++]AF.FF7$F
JF*FL7$FJ$\"1+++++++SFH7$$!\"#F*FN7$FQF*FS7$FQ$F7FH7$$!1++++++]<F.FU7$
FWF*FY7$FW$\"1*************z)FH7$$!1+++++++:F.Fen7$FhnF*Fjn7$Fhn$\"1++
++++?F!#;7$$!1++++++]7F.F\\o7$F`oF*Fbo7$F`o$\"1++++++g<F^o7$$!\"\"F*Fd
o7$FgoF*Fio7$Fgo$\"1++++++![#F^o7$$!1+++++++vF^oF[p7$F^pF*F`p7$F^p$\"1
++++++!o$F^o7$$!1+++++++]F^oFbp7$FepF*Fgp7$Fep$\"1+++++++WF^o7$$FCF^oF
ip7$F\\qF*F]q7$F\\qFbp7$F*Fbp7$F*F*F`q7$F*$\"1++++++?JF^o7$$\"1+++++++
DF^oFbq7$FeqF*Fgq7$Feq$\"1++++++!)GF^o7$$\"1+++++++]F^oFiq7$F\\rF*F^r7
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}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "N
ow let's compare the simulated mean to the theoretical mean.  " }}
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0 14 "theo_mean:=mu;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*theo_meanG
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