With the second, you have to first transform your random normals to remove even the random variation away from the zero mean and identity covariance (making the sample mean zero and sample covariance $I_n$), then proceed as before. Then $y$ has the desired population characteristics. a suitable $L$ could be obtained via Cholesky decomposition). If $\mu$ and $\Sigma$ are the population mean and covariance you need and $z$ are iid standard normal, you calculate $y=Lz+\mu$, for some $L$ where $LL'=\Sigma$ (e.g. With first case you could use random normals without the population structure (such as iid standard normal which have expectation 0 and identity covariance matrix) and then impose it - transform to get the covariance matrix and mean you want.
So for multivariate normals, either can be done in a fairly straightforward manner: You want the second case - but you get it by following the same approach as the first case, with an extra standardization step. Ii) you want to generate a sample whose sample characteristics match the ones specified (but, due to the constraints of exactly matching sample quantities to a prespecified set of values, don't really come from the distribution you want).
I) you want to generate a sample from a given distribution whose population characteristics match the ones specified (but due to sampling variation, you don't have the sample characteristics exactly matching). There are two different typical situations for these kind of problems:
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