Given a random variable $X$ with log normal distribution, can we find the probability of $X$ being greater than a positive constant $a$, i.e can we determine the integral $$ \int_a^\infty \frac{1}{xs\sqrt{2\pi}} e^{-(\ln(x)-s)^2/(2s^2)} dx $$ from a to infinity?
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The problem is equivalent to that of finding $Pr[Z\ge s^{-1}\log a] = \Phi(s^{-1}\log a)$ where $Z\sim N(0,1)$. There is no closed form expression for this value. Any statistical package will give you a numeric approximation. Also, there are analytic bounds of arbitrary sharpness available (ref)
mStudent
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Can you please name such a package? I actually want to find the value of a for which the integral is maximized. – adsj Jun 23 '13 at 17:59
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R-project is free and excellent. – mStudent Jun 23 '13 at 18:09
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if I understood you correctly, you want to maximize $a\mapsto Pr[X\ge a]$ where $X$ has log-normal distribution? – mStudent Jun 23 '13 at 18:10
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Yes, the actual problem I am working on is to find the value of a for which Pr(X>=a) is maximized. – adsj Jun 23 '13 at 18:12
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$Pr[X\ge a]$ is bounded above by $1$ and is monotone decreasing in $a$ on $(0,\infty)$. That is, there is no maximum. – mStudent Jun 23 '13 at 18:14
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Okay, thanks! But can you please prove your statement? – adsj Jun 23 '13 at 18:15
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Well, it's more of an axiom than a statement. The only fact we need to agree on, is that $Pr[Z<a]$ is strictly positive for any $a\in \mathbb{R}$ for $Z\sim N(0,1)$. – mStudent Jun 23 '13 at 18:16
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Yes I completely missed that. Thanks! – adsj Jun 23 '13 at 18:20