Pairwise vs. Mutual independence for normal random vector Lets say that I have a vector $X$ for which the marginal distribution of each element $x_1,x_2,...$ is normal with variance = 1. Additionally, assume that $X$ is pairwise independent. 
Does this prove that the covariance matrix of $X$ is the identity matrix? What is the difference in the distribution of $X$ if instead I assume that $X$ is mutually independent? Is the distribution any different?
Thanks,
Clark
 A: For many people, calling a vector a normal vector is equivalent to
assuming that the random variables in question are jointly normal
random variables. For others, such as yourself, that the
marginal density of each random variable is a normal density is
sufficient reason to call the vector a normal vector.
That being said, the covariance matrix doesn't care diddly-squat
as to whether the random variables are normal or not. The pairwise
independence of the (unit variance) random variables suffices to
ensure that the covariance matrix is the identity matrix regardless
of the actual multivariate distribution, and regardless of whether
the random variables are mutually independent or just pairwise
independent.  And no, pairwise independence of random
variables does not guarantee mutual independence of the
random variables even if the
random variables happen to be normal.  An example of three
pairwise independent standard normal random variables that
are not mutually independent random variables can be found
in the latter half of 
this answer of mine.
