Stochastic integral: $E\left(\int^1_0(W(s))\,ds\int^1_0t(W(t)\right)\,dt$

I need to calculate the expectation of the product between the integral of a Wiener process and the expectation of a Wiener process. Is the same as the expectation of the product between the integral and comprehensive Wiener by Wiener $t$. The integrals are evaluated between $0$ and $1$.

Necesito calcular la expectativa del producto entre la integral de un proceso de Wiener y la expectativa de un proceso de Wiener. Lo que es lo mismo la expectativa del producto entre la integral de Wiener y la integral de $t$ por Wiener. las integrales se evaluan entre $0$ y $1$.

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1 Answer

Hint: For every $s$ and $t$, $\mathbb E(W(s)W(t))=\min(s,t)$. Apply this to $$\mathbb E\left(\int_0^1W(s)\mathrm ds\cdot\int_0^1t\,W(t)\mathrm dt\right)=\int_0^1\!\!\!\int_0^1t\cdot\mathbb E\left(W(s)W(t)\right)\mathrm dt\mathrm ds.$$

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The answer is 5/24, you seem to know that (hence this seems to be (homework), and if it is you should add the tag). Did you try to start from my hint and to reach this answer? Let me suggest that you show what is stopping you. –  Did Oct 25 '12 at 17:18
You are not reading what I write nor answering to what I ask. What should I deduce from this attitude? Same question about your now deleted comment. –  Did Oct 25 '12 at 18:21
Two deleted comments now. Can we go back to the maths? –  Did Oct 26 '12 at 18:27