  In this paper, we study the power and limitations of computing effectively
generic sequences using effectively random oracles. Previously, it was known
that every 2-random sequence computes a 1-generic sequence (as shown by Kautz)
and every 2-random sequence forms a minimal pair in the Turing degrees with
every 2-generic sequence (as shown by Nies, Stephan, and Terwijn). We
strengthen these results by showing that every Demuth random sequence computes
a 1-generic sequence (which answers an open question posed by Barmpalias, Day,
and Lewis) and that every Demuth random sequence forms a minimal pair with
every pb-generic sequence (where pb-genericity is an effective notion of
genericity that is strictly between 1-genericity and 2-genericity). Moreover,
we prove that for every comeager $\mathcal{G}\subseteq 2^\omega$, there is some
weakly 2-random sequence $X$ that computes some $Y\in\mathcal{G}$, a result
that allows us to provide a fairly complete classification as to how various
notions of effective randomness interact in the Turing degrees with various
notions of effective genericity.
