# 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