In[1]:= f1[x_, μ_, σ_] := (E^(-((μ - Log[x])^2/(2 σ^2))) Sqrt[2/π])/(2 x σ)
Integrate[f1[x, μ, σ], x]
Out[1]= 1/2 Erf[(-μ + Log[x])/(Sqrt[2] σ)]
In[2]:=
p1 = {{0.3, 0.778}, {0.5, 0.912}, {1.0, 0.972}, {5.0, 0.998}};
q1 = ListPlot[p1, PlotStyle -> {Red, PointSize[Small]},
AxesLabel -> {"x", "y"}, PlotRange -> {0, 1.2}];
In[3]:=
g1[x_, μ_, σ_] := 1/2*Erf[(-μ + Log[x])/(Sqrt[2] σ)] + 0.5
nlm = NonlinearModelFit[p1, g1[x, μ, σ], {μ, σ}, x];
nlm["BestFitParameters"]
Out[3]= {μ -> -1.92447, σ -> 0.933737}
In[4]:= Show[q1,
Plot[g1[x, -1.92447, 0.933737], {x, 0, 5}, PlotRange -> {0, 1}]]
In[5]:= f2[x_, μ_, σ_] := x*f1[x, μ, σ]
Integrate[f2[x, μ, σ], x]
Out[5]= -(1/2) E^(μ + σ^2/2)
Erf[(μ + σ^2 - Log[x])/(Sqrt[2] σ)]
In[6]:=
p2 = {{0.02, 0.01}, {0.3, 0.125}, {0.5, 0.187}, {1.0, 0.234},
{5.0, 0.282}, {20.0, 0.302}};
q2 = ListPlot[p2, PlotStyle -> {Red, PointSize[Small]},
AxesLabel -> {"x", "y"}, PlotRange -> {-0.1, 0.35}];
In[7]:= g2[x_, μ_, σ_, λ_] := -(1/2) E^(μ + σ^2/2)
Erf[(μ + σ^2 - Log[x])/( Sqrt[2] σ)] + λ
nlm = NonlinearModelFit[p2, g2[x, μ, σ, λ], {μ, σ, λ}, x];
nlm["BestFitParameters"]
Out[7]= {μ -> -1.72573, σ -> 0.914071, λ -> 0.154702}
In[8]:= Show[q2,
Plot[g2[x, -1.72573, 0.914071, 0.154702], {x, 0, 20},
PlotRange -> {0, 0.35}]]