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I want to find the joint distribution of the random vector $(W_t, \int_0^t W_s \; \mathrm ds)$

where $W_t$ is Brownian motion. I know $W_t \sim N(0,t)$, but I don't know how to calculate the distribution of the integral

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    In this question http://math.stackexchange.com/questions/27841/how-to-calculate-e-int-0tw-sdsn-n-geq-2 it was shown that the second coordinate of your vector is a Gaussian Process, so this two Gaussian Process have multivariated normal distribution. See the link for more details.2011-10-22
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    @Leandro, *the second coordinate of your vector is a Gaussian Process, so this two Gaussian Process have multivariated normal distribution*... No, the distributions of the coordinate processes are not enough.2011-12-16

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