= Solution
For each substudy define the log-odds treatment effect
$$
d_j=\theta_{1j}
=\operatorname{logit}(p_{j1})-\operatorname{logit}(p_{j0}).
$$
The normal random-effects log-likelihood, up to an additive constant, is
$$
\ell(\mu,\sigma^2)
=-\frac J2\log\sigma^2
-\frac1{2\sigma^2}\sum_{j=1}^J(d_j-\mu)^2.
$$
Its <score equations> give
$$
\widehat\mu=\frac1J\sum_{j=1}^Jd_j,
\qquad
\widehat\sigma^2=\frac1J\sum_{j=1}^J(d_j-\widehat\mu)^2.
$$
These are the <maximum-likelihood estimators> rather than the unbiased sample-variance estimator. At an interior solution with $\widehat\sigma^2>0$, the Hessian in $(\mu,\sigma^2)$ is negative definite, which is the required second-order condition.
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