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January 30th, 2017, 12:03 PM  #1 
Senior Member Joined: Feb 2015 From: london Posts: 121 Thanks: 0  Expectation proof
$\displaystyle E[e^x1] = \Phi (\frac{u +t}{\sqrt(t)}) e^{u+t/2}  \Phi (\frac{u}{\sqrt(t)})$ where $\displaystyle \Phi$ deontes the cumulative distribution function of the N(0,1) Normal distribution i.e $\displaystyle \Phi(x) = (2\pi)^{0.5} \int_{\infty}^{x} e^{\frac{s^2}{2} } ds$ prove that: $\displaystyle E[e^x 1 ]  E[1  e^x] = e^{u+t/2}  1$ Any tips on how to prove this. Every time I try, I seem to just end up with $\displaystyle 2E[e^x 1]$ 
January 30th, 2017, 05:26 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,258 Thanks: 508 
Something is wrong in your statement. It looks like you are doubling the original expression.

January 31st, 2017, 04:38 AM  #3 
Senior Member Joined: Feb 2015 From: london Posts: 121 Thanks: 0 
Which statement do you think is wrong?

January 31st, 2017, 07:36 AM  #4 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 851 Thanks: 308 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  

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