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Say I have a $\text{discrete}$ multivariate random variable $X=[X_1,X_2,\ldots,X_n]$, where each $X_i$ is of the same distribution class. Define random variable $Y$ to be the Manhattan distance between two samples of $X$, and define $Z$ to be the 'normal' Euclidean distance.

Obviously, $Y$ and $Z$ are positively correlated. Given the parameters of the distribution of $X$, is it possible to express this correlation? For instance, given $n$ dimensions, let each $X_i$ be an independent Bernoulli with parameter $p_i$.

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Well, for Bernoulli random variables, even in the non-independent case, $Y=Z^2$. This should simplify the computations. –  D. Thomine Jul 8 '12 at 20:35
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