# Distribution of integral of a normally distributed random variable

What can we say about distribution of

$\int_t^TN(\mu(s),\sigma^2(s))ds$

,where $N(\mu,\sigma)$ is independent normally distributed with mean $\mu(s)$ and variance $\sigma^2(s)$, $T$ and $t$ are finite?

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To even define the integral $\int\limits_t^TX_sds$ where $X_s$ has gaussian distribution $(\mu(s),\sigma^2(s))$, one needs to specify the joint distribution of the family $(X_s)_{t\leqslant s\leqslant T}$. –  Did Oct 27 '12 at 13:07
@did Yes, I forgot this. Let's say they are iid. –  learningmath Oct 27 '12 at 22:48
Then the integral will not be defined. –  Did Oct 27 '12 at 22:53
They can't be iid; you're explicitly saying that they have different distributions (different means and variances) -- @did you mean "independently distributed", as the question now reads? –  joriki Oct 27 '12 at 23:07
@joriki Yes, sorry. I meant just indepedent. –  learningmath Oct 27 '12 at 23:11

I think the object you really want to be integrating is a Gaussian process with some specified mean and covariance function. And you'll probably want the covariance function to have some regularity to it (e.g. continuous) in order to guarantee that the integral exists. In this case the result of the integral will be normally distributed, so you only need to compute its mean and variance. This is straightforward using Fubini's theorem. See also my answer here.

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