= Solution
Under <independence>, the log <likelihood function> separates:
$$
\ell(\alpha,\beta)=\sum_i n_{i.}\log\alpha_i+
\sum_j n_{.j}\log\beta_j.
$$
Maximizing the two terms over their respective <probability> simplices, by the same <Lagrange multiplier> argument as in part (a), gives
$$
\widehat\alpha_i=\frac{n_{i.}}N,\qquad
\widehat\beta_j=\frac{n_{.j}}N.
$$
Therefore the constrained <maximum likelihood estimation>[maximum-likelihood estimates] are
$$
\boxed{\widehat\theta_{ij}^{\,0}=\frac{n_{i.}n_{.j}}{N^2}.}
$$
The associated fitted cell counts are $E_{ij}=N\widehat\theta_{ij}^{\,0}=n_{i.}n_{.j}/N$.
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