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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The mean exit time of the \+
process out of the interval [0," }{TEXT 256 1 "a" }{TEXT -1 43 "] is g
overned by the boundary value problem" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "\{m*diff(mu(x),x)+s^2*diff(mu(x),x$2)/2=-1,mu(0)=0,mu
(a)=0\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/,&*&%\"mG\"\"\"-%%diffG
6$-%#muG6#%\"xGF/F(F(*&#F(\"\"#F(*&)%\"sGF2F(-F*6$F,-%\"$G6$F/F2F(F(F(
!\"\"/-F-6#\"\"!F?/-F-6#%\"aGF?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
15 "The solution is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "R9 :
= rhs(dsolve(\{m*diff(mu(x),x)+s^2*diff(mu(x),x$2)/2=-1,mu(0)=0,mu(a)=
0\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R9G,(**%\"aG\"\"\",&-%$ex
pG6#,$**\"\"#F(%\"mGF(%\"sG!\"#F'F(!\"\"F(F(F3F3F0F3-F+6#,$**F/F(F0F(F
1F2%\"xGF(F3F(F(*&F0F3F8F(F3*(F0F3F'F(F)F3F3" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 70 "In our case the process starts in the cen
ter of the interval, i.e. at " }{TEXT 257 1 "a" }{TEXT -1 61 "/2. We t
herefore find the mean exit time of the process to be" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(R9,x=a/2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,(**%\"aG\"\"\",&-%$expG6#,$**\"\"#F&%\"mGF&%\"sG!\"#
F%F&!\"\"F&F&F1F1F.F1-F)6#,$*(F.F&F/F0F%F&F1F&F&*(F-F1F.F1F%F&F1*(F.F1
F%F&F'F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "which is simplified
 to" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M0 := simplify(%);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M0G,$*&#\"\"\"\"\"#F(**,&-%$expG6
#,$*(%\"mGF(%\"sG!\"#%\"aGF(!\"\"F(F(F5F(F4F(F1F5,&F,F(F(F(F5F(F5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "The second order moment of the exi
t time, nu(x), is governed by the equation" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 53 "ode := m*diff(nu(x),x)+s^2*diff(nu(x),x$2)/2 = -2*R
9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,&*&%\"mG\"\"\"-%%diffG6
$-%#nuG6#%\"xGF0F)F)*&#F)\"\"#F)*&)%\"sGF3F)-F+6$F--%\"$G6$F0F3F)F)F),
(*,F3F)%\"aGF),&-%$expG6#,$**F3F)F(F)F6!\"#F>F)!\"\"F)F)FFFFF(FF-FA6#,
$**F3F)F(F)F6FEF0F)FFF)FF*(F3F)F(FFF0F)F)**F3F)F(FFF>F)F?FFF)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Subject to the boundary nu(0) = n
u(a) = 0, nu(x) is solved for, and evaluated at our starting point " }
{TEXT 258 1 "a" }{TEXT -1 2 "/2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 30 "dsolve(\{ode,nu(a)=0,nu(0)=0\}):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "eval(%,x=a/2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
31 "The variance is defined by E\{T^" }{XPPEDIT 18 0 "2" "6#\"\"#" }
{TEXT -1 30 "\}- E\{T\}^2 = nu(a/2)-mu(a/2)^2:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "V := rhs(%)-eval(R9,x=a/2)^2:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "R0 := simplify(V):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 60 "The case that the drift rate is zero is of special interest:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "limit(R0,m=0);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*(\"#C!\"\"%\"aG\"\"%%\"sG!\"%\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 12 "" 1 "" 
{TEXT -1 59 "The original expression can be greatly simplified by Mapl
e:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "expand(R0):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "V1 := simplify(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#V1G,$*&#\"\"\"\"\"#F(**%\"aGF(,(*&-%$expG
6#,$**F)F(%\"mGF(%\"sG!\"#F+F(F(F()F4F)F(!\"\"**F)F(F3F(F+F(-F/6#*(F3F
(F4F5F+F(F(F(*$F6F(F(F(F3!\"$,&F(F(F9F(F5F(F7" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 44 "To tidy the presentation the parameter y = -" }{TEXT 
261 1 "m" }{TEXT -1 1 " " }{TEXT 260 1 "a" }{TEXT -1 3 " / " }{TEXT 
259 1 "s" }{TEXT -1 17 "^2 is introduced:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "subs(\{-m*a/s^2=y,m*a/s^2=-y\},V1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&**%\"aGF&,(*&-%$expG6#,$*&F'F&%
\"yGF&!\"\"F&)%\"sGF'F&F2**F'F&%\"mGF&F)F&-F-6#,$F1F2F&F&*$F3F&F&F&F6!
\"$,&F&F&F7F&!\"#F&F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 256 "" 1 "" {TEXT -1 124 "To investigate the relation b
etween the mean and variance, we wish to express the variance as a rat
io of the expected value:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 
"V1/M0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,(*&-%$expG6#,$**\"\"#\"
\"\"%\"mGF,%\"sG!\"#%\"aGF,F,F,)F.F+F,!\"\"**F+F,F-F,F0F,-F'6#*(F-F,F.
F/F0F,F,F,*$F1F,F,F,F-F/,&F,F,F4F,F/,&-F'6#,$F6F2F,F,F2F2,&F:F,F,F,F,
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rat := simplify(%);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$ratG,$*(,(*&-%$expG6#,$**\"\"#\"\"\"%\"mGF/%\"s
G!\"#%\"aGF/F/F/)F1F.F/!\"\"**F.F/F0F/F3F/-F*6#*(F0F/F1F2F3F/F/F/*$F4F
/F/F/F0F2,&F/F5F)F/F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "
subs(\{m*a/s^2=-y\},%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,(*&-%$
expG6#,$*&\"\"#\"\"\"%\"yGF-!\"\"F-)%\"sGF,F-F/**F,F-%\"mGF-%\"aGF--F(
6#,$F.F/F-F-*$F0F-F-F-F3!\"#,&F-F/F'F-F/F/" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 129 "[ To convert the Maple presentation to the same previous
 presentation, we multiply the above expression by 1 = exp(2y)/exp(2y)
: ]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "simplify(expand(simp
lify(expand(%*exp(2*y)))/exp(2*y)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#*(,(*$)%\"sG\"\"#\"\"\"!\"\"**F(F)%\"mGF)%\"aGF)-%$expG6#%\"yGF)F)*&
-F/6#,$*&F(F)F1F)F)F)F&F)F)F)F,!\"#,&F)F*F3F)F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "limit(V1,m=0)/limit(M0,m=0);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,$*(\"\"'!\"\"%\"sG!\"#%\"aG\"\"#\"\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 154 "[ We similarly convert the Maple present
ation of the variance to a previously derived presentation, i.e., we m
ultiply the expression for the variance also" }}{PARA 0 "" 0 "" {TEXT 
-1 26 " by 1 = exp(2y)/exp(2y): ]" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "V1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&**%\"aGF&,(*&-%$expG6#,$**F'F&%\"mGF&
%\"sG!\"#F)F&F&F&)F2F'F&!\"\"**F'F&F1F&F)F&-F-6#*(F1F&F2F3F)F&F&F&*$F4
F&F&F&F1!\"$,&F&F&F7F&F3F&F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "subs(m*a/s^2=-y,V1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"
\"\"\"\"#F&**%\"aGF&,(*&-%$expG6#,$*&F'F&%\"yGF&!\"\"F&)%\"sGF'F&F2**F
'F&%\"mGF&F)F&-F-6#,$F1F2F&F&*$F3F&F&F&F6!\"$,&F&F&F7F&!\"#F&F2" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "simplify(expand(simplify(exp
and(%*exp(2*y)))/exp(2*y)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#
\"\"\"\"\"#F&**%\"aGF&,(*$)%\"sGF'F&!\"\"**F'F&%\"mGF&F)F&-%$expG6#%\"
yGF&F&*&-F26#,$*&F'F&F4F&F&F&F,F&F&F&F0!\"$,&F1F&F&F&!\"#F&F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "limit(M0,m=0);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$*(\"\"%!\"\"%\"sG!\"#%\"aG\"\"#\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "rat;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*(,(*&-%$expG6#,$**\"\"#\"\"\"%\"mGF-%\"sG!\"#%\"aGF-
F-F-)F/F,F-!\"\"**F,F-F.F-F1F--F(6#*(F.F-F/F0F1F-F-F-*$F2F-F-F-F.F0,&F
-F3F'F-F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "We investigate th
e relation between the standard deviation of T [= sqrt(var(T))] and th
e mean: We compute a Taylor expansion for the `coefficient of variatio
n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "series(sqrt(V1)/M0,m
,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"mG,$*,\"\"$!\"\"\"\"'#\"
\"\"\"\"#*&%\"aG\"\"%%\"sG!\"%F*F0F,F.!\"#F+\"\"!,$**\"$!=F(F)F*F-F*F0
F2F(F,-%\"OG6#F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simpl
ify(%,'assume=real');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"mG,$*&\"
\"$!\"\"\"\"'#\"\"\"\"\"#F+\"\"!,$**\"$!=F(F)F*%\"aGF,%\"sG!\"%F(F,-%
\"OG6#F+\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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