Suppose that (left{mu_{i j} ight}) satisfy a multiplicative model [begin{equation*}mu_{i j}=mu alpha_{i} beta_{j}, quad 1 leq i

Suppose that \(\left\{\mu_{i j}\right\}\) satisfy a multiplicative model

\[\begin{equation*}\mu_{i j}=\mu \alpha_{i} \beta_{j}, \quad 1 \leq i \leq I, 1 \leq j \leq J \tag{7.30}\end{equation*}\]

where \(\left\{\alpha_{i}, i=1, \ldots, I\right\}\) and \(\left\{\beta_{j}, j=1, \ldots, J\right\}\) are positive numbers satisfying the constraint

\[\sum_{i=1}^{I} \alpha_{i}=\sum_{j=1}^{J} \beta_{j}=1\]

(a) Compute the multinomial distribution conditional on \(\sum_{i=1}^{I} \sum_{j=1}^{J} n_{i j}=n\), and verify that \(\alpha_{i}\) and \(\beta_{j}\) are actually the marginal probabilities of the row and column variables of the two-way contingency table, respectively.

(b) Prove that the row and column variables are independent.