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I am trying to work out the following two integrals involving a standard Brownian motion started at $W_0 = 0$.

The first expression is bewildering me a bit, since it seems like somehow the Itô isometry is involved and would help me work the expectation. $$ \mathbb E\left[\left(\frac{1}{T}\int_0^T\,W_t\,dt \right)^2\right] $$ Secondly, it seems like to prove the following inequality it could be useful to reach an expression resulting in $\frac{2}{\sqrt{\pi}}$, but I am not sure how to do it. $$ \mathbb P\left(\int_0^1\,W_t\,dt > \frac{2}{\sqrt{3}} \right) $$

Any help is welcome!

Adam
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  • @ Adam : Think about Integration by Parts formula, then you should arrive to a sum of Gaussian unless mistaken (I haven't the calculations though too lazy for that). Best regards – TheBridge Jan 19 '15 at 21:46
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    @TheBridge Sorry but this is muuuch more direct. – Did Jan 20 '15 at 00:29
  • @ Did : No need to be sorry, if it has already been solved in an old post then everything is fine. Best regards – TheBridge Jan 20 '15 at 07:57

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