= Solution
At stage 1, the <maximum-likelihood estimators> of the two normal arm means are their sample means. Their difference estimates the treatment effect:
$$
\widehat\delta_1=Y_{11}-Y_{01}.
$$
Each arm has $n_1$ patients and <variance> $\sigma^2/n_1$, so <independence> across arms gives
$$
\boxed{\widehat\delta_1\sim N\left(\delta,\frac{2\sigma^2}{n_1}\right)
=N(\delta,I_1^{-1}),\qquad I_1=\frac{n_1}{2\sigma^2}.}
$$
The quantity $I_1$ is the <Fisher information> for the mean difference when the common <variance> is known. The PDF's preliminary expression uses $n_i$ for the <variance> denominator, although $i$ indexes arms and $n_0$ is not defined. The stage-specific <sample size> is $n_j$; the calculation uses this intended reading, consistent with the specified $n_1,n_2$ per-arm recruitment and the later weighted <estimator>. Fresh stage-2 patients are independent of stage-1 patients in the usual trial model; the cumulative stage means themselves are not independent across stages.
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