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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Q1:  restart:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "F:=1-exp(-x^theta);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,&\"\"\"F&-%$expG6#,$)%\"xG%&the
taG!\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=diff(F,x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG**)%\"xG%&thetaG\"\"\"F(F)F'
!\"\"-%$expG6#,$F&F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f
2d:=simplify(diff(ln(f),theta,theta));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$f2dG,$*&,&*()%&thetaG\"\"#\"\"\")-%#lnG6#%\"xGF+F,)F1F*F,F,F,
F,F,F*!\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(a
>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "simplify(int(f2d*f,
x=0..infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"'!\"\",*F%
\"\"\"*$)%#PiG\"\"#F(F(*&\"#7F(%&gammaGF(F&*&F%F()F/F,F(F(F(%&thetaG!
\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(-1/%/n);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*($\"+fhT$[&!#5\"\"\"%&thetaG\"\"#%
\"nG!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Q2: restart:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:=(1+2*theta*x)/(1+the
ta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,&\"\"\"F'*(\"\"#F'%&t
hetaGF'%\"xGF'F'F',&F'F'F*F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "mu:=simplify(int(x*f,x=0..1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#muG,$*(\"\"'!\"\",&*&\"\"%\"\"\"%&thetaGF,F,\"\"$F,F
,,&F,F,F-F,F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "est:=sol
ve(mu=Xbar,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$estG,$**\"\"$
\"\"\"\"\"#!\"\",&F(F**&F)F(%%XbarGF(F(F(,&F)F**&F'F(F-F(F(F*F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "X:=[0.76, 0.50, 0.02, 0.19, \+
0.77, 0.28, 0.79, 0.29, 0.99, 0.63, 0.84, 0.67];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"XG7.$\"#w!\"#$\"#]F($\"\"#F($\"#>F($\"#xF($\"#GF($
\"#zF($\"#HF($\"#**F($\"#jF($\"#%)F($\"#nF(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 37 "Xbar:=sum(X[i],i=1..nops(X))/nops(X);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%%XbarG$\"+LLL3c!#5" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 4 "est;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7:.[d!#
5" }}}{EXCHG {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "Q3:  restart:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "f:=sqrt(theta)/Pi/(1+theta*x^2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG*(%&thetaG#\"\"\"\"\"#%#PiG!\"\",&F(F(*&F&F()%\"x
GF)F(F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "lnf:=ln(theta)
/2-ln(Pi)-ln(1+theta*X[i]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ln
fG,(*&#\"\"\"\"\"#F(-%#lnG6#%&thetaGF(F(-F+6#%#PiG!\"\"-F+6#,&F(F(*&F-
F()&%\"XG6#%\"iGF)F(F(F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 
"eq:=sum(diff(lnf,theta),i=1..n)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#eqG/,&*(\"\"#!\"\"%\"nG\"\"\"%&thetaGF)F+-%$sumG6$,$*&&%\"XG6#%\"i
GF(,&F+F+*&F,F+)F2F(F+F+F)F)/F5;F+F*F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 61 "X:=[-0.90,0.32,-3.47,-4.84,2.47,7.58,2.48,-3.28, 5
.24, 4.03];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG7,$!#!*!\"#$\"#KF
($!$Z$F($!$%[F($\"$Z#F($\"$e(F($\"$[#F($!$G$F($\"$C&F($\"$.%F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "n:=nops(X);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "fsolve(eq,theta=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#$\"+wlb36!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Q4: rest
art:  assume(theta>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f
:=exp(-(ln(x)^2)/theta)/x/sqrt(Pi*theta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG*(-%$expG6#,$*&-%#lnG6#%\"xG\"\"#%'theta|irG!\"\"
F1\"\"\"F.F1*&%#PiGF2F0F2#F1F/" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "product(subs(x=X[i],f)
,i=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%(productG6$*(-%$expG6#,
$*&-%#lnG6#&%\"XG6#%\"iG\"\"#%'theta|irG!\"\"F5\"\"\"F/F5*&%#PiGF6F4F6
#F5F3/F2;F6%\"nG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "This can be r
e-written as a product of:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 75 "exp(-sum(ln(X[i])^2,i=1..n)/theta)/(Pi*theta)^(n
/2),1/product(X[i],i=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&-%$ex
pG6#,$*&-%$sumG6$*$)-%#lnG6#&%\"XG6#%\"iG\"\"#\"\"\"/F4;F6%\"nGF6%&the
taG!\"\"F;F6)*&%#PiGF6F:F6,$F9#F6F5F;*&F6F6-%(productG6$F1F7F;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Implying that a sufficient statist
ic for estimating theta is:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "sum(ln(X[i])^2,i=1..n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$sumG6$*$)-%#lnG6#&%\"XG6#%\"iG\"\"#\"\"\"/F.;F0%\"nG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Its expected value is  n*   t
he expected value of" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "simplify(int(ln(x)^2*f,x=0..infinity));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"%'theta|irG\"\"\"F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 13 "implying that" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "est:=2*sum(ln(X[i])^2,i=1..n
)/n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$estG,$*(\"\"#\"\"\"-%$sumG6
$*$)-%#lnG6#&%\"XG6#%\"iGF'F(/F4;F(%\"nGF(F7!\"\"F(" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 34 "is an unbiased estimator of theta." }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "X:=[0.91, 0.79, \+
0.82, 0.97, 1.08, 1.39, 0.40, 1.34];" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"XG7*$\"#\"*!\"#$\"#zF($\"##)F($\"#(*F($\"$3\"F($\"$R\"F($\"#SF($
\"$M\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "n:=nops(X);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\")" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 4 "est;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+sb%4'G!
#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Q5:  restart:  assume
(b>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f:=exp(-(x-a)/b)/
(1+exp(-(x-a)/b))^2/b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(-%$e
xpG6#,$*&,&%\"xG\"\"\"%\"aG!\"\"F-%#b|irGF/F/F-,&F-F-F&F-!\"#F0F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "mu:=simplify(int(x*f,x=-infi
nity..infinity),symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#muG%
\"aG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "sigma:=simplify(sqr
t(int((x-mu)^2*f,x=-infinity..infinity)),symbolic);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&sigmaG,$**\"\"$!\"\"F'#\"\"\"\"\"#%#PiGF*%#b|irGF*
F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(\{mu=Xbar,sigma
=S\},\{a,b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"aG%%XbarG/%#b|
irG*(%\"SG\"\"\"\"\"$#F+\"\"#%#PiG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "X:=[-3.4, -5.7, -5.2, -2.5, -0.8, 3.4, -19.0, 2.8, 0.
1, 3.4, -7.2, -6.0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG7.$!#M!
\"\"$!#dF($!#_F($!#DF($!\")F($\"#MF($!$!>F($\"#GF($\"\"\"F(F1$!#sF($!#
gF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "'a' estimate:" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sum(X[i],i=1..12
)/12.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+nmmTL!\"*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 13 "'b' estimate:" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "evalf(sqrt(3)/Pi*sqrt(sum((X
[i]-Xbar)^2,i=1..12)/12.));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lfQ
wK!\"*" }}}}{MARK "48" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }