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January 30th, 2017, 12:03 PM   #1
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Expectation proof

$\displaystyle E[e^x-1] = \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 $\displaystyle 2E[e^x -1]$
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January 30th, 2017, 05:26 PM   #2
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Something is wrong in your statement. It looks like you are doubling the original expression.
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January 31st, 2017, 04:38 AM   #3
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Which statement do you think is wrong?
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January 31st, 2017, 07:36 AM   #4
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Quote:
Originally Posted by calypso View Post
Which statement do you think is wrong?
The last seems slightly odd: $E(e^X - 1) - E(1 - e^X) = E(e^X - 1) + E(e^X - 1) = 2E(e^X - 1)$.
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