= Solution
Under the independent-gene version of the model, integrating both component labels and effects gives
$$
\boxed{p(y\mid q,V)=\prod_{i=1}^N\{(1-q)f_0(y_i)+qf_1(y_i)\}.}
$$
This product uses independence of the gene effects and observation errors conditional on the shared parameters. Maximize its logarithm over $0\leq q\leq1$, for example with a one-dimensional bounded optimizer or a grid followed by refinement. Its derivatives are
$$
\ell'(q)=\sum_i\frac{f_1(y_i)-f_0(y_i)}{(1-q)f_0(y_i)+qf_1(y_i)},
\qquad
\ell''(q)=-\sum_i\frac{[f_1(y_i)-f_0(y_i)]^2}{[(1-q)f_0(y_i)+qf_1(y_i)]^2}\leq0.
$$
Thus the log <likelihood function> is concave. An interior <maximum-likelihood estimate> solves the score equation; otherwise it is at zero or one, determined by the endpoint score signs. An alternative <expectation-maximization algorithm> iterates
$$
w_i=\frac{qf_1(y_i)}{(1-q)f_0(y_i)+qf_1(y_i)},\qquad
q_{\mathrm{new}}=\frac1N\sum_iw_i.
$$
For $V=0$, the components coincide and $q$ is not identifiable; for $V>0$ the full observation values carry information about the mixing fraction.
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