We consider $\alpha = (\alpha_i)_{i \in I} \in (\mathbb{R}_+^*)^I$ and $\beta = (\beta_j)_{j \in J} \in (\mathbb{R}_+^*)^J$ such that $\sum_{i \in I} \alpha_i = \sum_{j \in J} \beta_j = 1$. We denote $$F(\alpha, \beta) = \left\{q \in Q \mid \sum_{j' \in J} q_{ij'} = \alpha_i \text{ and } \sum_{i' \in I} q_{i'j} = \beta_j \text{ for all } (i,j) \in I \times J\right\}.$$ We denote by $\boldsymbol{p}$ the element of $F(\alpha, \beta)$ defined by $p_{ij} = \alpha_i \beta_j > 0$ for all $(i,j) \in I \times J$. Let $X_1$ and $X_2$ be two random variables such that $X_1$ takes values in $I$ and $X_2$ takes values in $J$. (a) Verify that if $\boldsymbol{q} \in F(\alpha, \beta)$, then $\sum_{i \in I} \sum_{j \in J} q_{ij} = 1$. (b) Assume that $P(X_1 = i, X_2 = j) = q_{ij}$ with $q \in F(\alpha, \beta)$. Calculate the distribution of $X_1$ and that of $X_2$ in terms of $\alpha$ and $\beta$. (c) What can we say about $X_1$ and $X_2$ when $\boldsymbol{q} = \boldsymbol{p}$?
We consider $\alpha = (\alpha_i)_{i \in I} \in (\mathbb{R}_+^*)^I$ and $\beta = (\beta_j)_{j \in J} \in (\mathbb{R}_+^*)^J$ such that $\sum_{i \in I} \alpha_i = \sum_{j \in J} \beta_j = 1$. We denote
$$F(\alpha, \beta) = \left\{q \in Q \mid \sum_{j' \in J} q_{ij'} = \alpha_i \text{ and } \sum_{i' \in I} q_{i'j} = \beta_j \text{ for all } (i,j) \in I \times J\right\}.$$
We denote by $\boldsymbol{p}$ the element of $F(\alpha, \beta)$ defined by $p_{ij} = \alpha_i \beta_j > 0$ for all $(i,j) \in I \times J$.\\
Let $X_1$ and $X_2$ be two random variables such that $X_1$ takes values in $I$ and $X_2$ takes values in $J$.\\
(a) Verify that if $\boldsymbol{q} \in F(\alpha, \beta)$, then $\sum_{i \in I} \sum_{j \in J} q_{ij} = 1$.\\
(b) Assume that $P(X_1 = i, X_2 = j) = q_{ij}$ with $q \in F(\alpha, \beta)$. Calculate the distribution of $X_1$ and that of $X_2$ in terms of $\alpha$ and $\beta$.\\
(c) What can we say about $X_1$ and $X_2$ when $\boldsymbol{q} = \boldsymbol{p}$?