Joint expectation of normal distribution I have a vector $v=(v_1,...,v_J)$, which is jointly normal. I then need to take the expectation of this vector. This should give me something like this:
1)$$E(v)=(E(v_1),...,E(v_J))$$
However, I want to write up the integral for this expectation, but I'm unsure how to do this. My guess is something like this:
$$E(v)=\int v \phi(v)dv$$
How does this relate to the expectation I have in 1)? $\phi(.)$ is the joint pdf, but to compute something like $(E(v_1),...,E(v_J))=$, I guess I need the marginal distribution: 
$$E(v_1,...,v_J)=(\int v_1 \phi(v_1),\int v_2 \phi(v_2),...,\int v_J \phi(v_J))$$  
Thanks
 A: You don't need the marginal distributions of the $J$ random variables to
compute the expectation of your vector; LOTUS allows you to compute the
means directly from the joint distribution (with the marginal actually
being computed along the way).  With just two variables (for simplicity),
LOTUS gives
$$\begin{align}
E[X] 
&= \int_{-\infty}^\infty \int_{-\infty}^\infty x\, f_{X,Y}(x,y)\, \mathrm dy
\,\mathrm dx & (\text{LOTUS)}\\
&= \int_{-\infty}^\infty x\, \left[\int_{-\infty}^\infty f_{X,Y}(x,y)
\, \mathrm dy \right] \,\mathrm dy &\text{(write it as iterated integral)}\\
&= \int_{-\infty}^\infty x \,f_{X}(x) \, \mathrm dx
&\text{(inner integral gives marginal density)}
\end{align}$$
Note that this applies even if the random variables are not jointly normal.
Of course, for jointly normal random variables, the means can be obtained
directly from the density itself in the sense that the usual specification
of a jointly normal density function is in terms of the mean vector
(which is what you are looking for) and the covariance matrix.
