Expression 8: "a" equals 1 half StartRoot, 1 minus 3ln left parenthesis, StartFraction, "s" Over "m" , EndFraction , right parenthesis minus StartRoot, left parenthesis, ln StartFraction, "s" Over "m" , EndFraction minus 3 , right parenthesis squared minus 8 , EndRoot , EndRoota=121−3lnsm−lnsm−32−8
equals=
0.4 1 9 7 3 5 9 0 0 6 3 40.419735900634
8
Log-Normal CDF:
9
Expression 10: "C" left parenthesis, "x" , right parenthesis equals 1 half ( 1 plus "E" Subscript, "r" "f" , Baseline ( StartFraction, ln "x" minus "b" Over "a" StartRoot, 2 , EndRoot , EndFraction ) )Cx=12(1+Erf(lnx−ba2))
10
Complementary Log-Normal CDF:
11
Expression 12: "C" Subscript, "C" , Baseline left parenthesis, "x" , right parenthesis equals 1 minus "C" left parenthesis, "x" , right parenthesisCCx=1−Cx
12
erf() using Abramowitz and Stegun formula 7.1.26 (error < 1.5e-7)
13
Expression 14: "E" Subscript, "r" "f" , Baseline left parenthesis, "x" , right parenthesis equals signum left parenthesis, "x" , right parenthesis left parenthesis, 1 minus left parenthesis, left parenthesis, left parenthesis, left parenthesis, left parenthesis, 1.0 6 1 4 0 5 4 2 9 times "t" Subscript, "A" "S" , Baseline left parenthesis, "x" , right parenthesis plus negative 1.4 5 3 1 5 2 0 2 7 , right parenthesis times "t" Subscript, "A" "S" , Baseline left parenthesis, "x" , right parenthesis , right parenthesis plus 1.4 2 1 4 1 3 7 4 1 , right parenthesis times "t" Subscript, "A" "S" , Baseline left parenthesis, "x" , right parenthesis plus negative 0.2 8 4 4 9 6 7 3 6 , right parenthesis times "t" Subscript, "A" "S" , Baseline left parenthesis, "x" , right parenthesis plus 0.2 5 4 8 2 9 5 9 2 , right parenthesis times "t" Subscript, "A" "S" , Baseline left parenthesis, "x" , right parenthesis times exp left parenthesis, negative left parenthesis, "x" times "x" , right parenthesis , right parenthesis , right parenthesisErfx=signx1−1.061405429·tASx+−1.453152027·tASx+1.421413741·tASx+−0.284496736·tASx+0.254829592·tASx·exp−x·x
14
Expression 15: "t" Subscript, "A" "S" , Baseline left parenthesis, "x" , right parenthesis equals StartFraction, 1 Over 1 plus 0.3 2 7 5 9 1 1abs left parenthesis, "x" , right parenthesis , EndFractiontASx=11+0.3275911absx