How to find expected value of normal cdf with a lognormal variable?
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Hi. I am trying to find E[N((a+(si^2)/2)/si)] where a is a constant, si is lognormally distributed with mean m and variance v, (m and v are basically constants) and N is the standard normal cdf. Any help is greatly appreciated...
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Answers (2)
Torsten
on 28 Oct 2014
MATHEMATICA cannot find an analytical expression for the corresponding integral.
So I guess you will have to assume numerical values for m and v.
Best wishes
Torsten.
Torsten
on 28 Oct 2014
Set values for s and m and enter the integral under
with
0.5*(1+Erf[x/Sqrt[2]])/(Sqrt[2*Pi]*s*x)*Exp[-(Log[x]-m)^2/(2*s^2)]
as integrand.
Best wishes
Torsten.
3 Comments
Torsten
on 28 Oct 2014
I don't understand what you are trying to do.
If g: IR -> IR is measurable,the expected value of g(X) is given by integral_{x=-oo}^{x=+oo} g(x)*f(x) dx where f is the density function for the random variable X. In your case,
g(x)=0.5*(1+Erf((a+x^2/2)/x/sqrt(2))) and
f(x)=density function of the lognormal distribution.
(I assumed you mean the lognormal distribution with parameters m and v^2, not mean m and variance v^2. If you really mean mean m and variance v^2, you will have to calculate its parameters m' and v'^2 in advance.)
Best wishes
Torsten.
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