# Distribution of merged random sets

Let $$A$$ and $$B$$ be finite disjoint sets of cardinality $$N$$. Suppose $$X_a,X_b,X_{a\to b}, X_{b\to a}$$, where $$(a,b)\in A\times B$$, are independent Bernoulli random variables so that $$X_a,X_b\sim\mathcal{B}(p)$$, $$X_{a\to b},X_{b\to a}\sim\mathcal{B}(q)$$ for parameters $$0\leq p,q\leq 1$$.

Create random subsets $$T_A\subset A$$ and $$T_B\subset B$$ by setting $$T_{A}=\{a\in A\mid X_a=1\}$$, $$T_{B}=\{b\in B\mid X_b=1\}$$. Thus, $$\#T_A$$ and $$\#T_B$$ follow the binomial distribution $$\mathcal{B}(N,p)$$. Let $$C=A\sqcup B$$ and create a random subset $$T_C\subset C$$ like so: $$\begin{cases} \forall a\in A,~a\in T_C\iff a\in T_A\text{ and }\exists b\in T_B,X_{b\to a} = 1 \\ \\ \forall b\in B,~b\in T_C\iff b\in T_B\text{ and }\exists a\in T_A,X_{a\to b} = 1\end{cases}$$ A computation shows that the probability of $$c\in C$$ lying in $$T_C$$ is $$p'=p(1-(1-pq)^N)$$, but the events $$[c\in T_C]$$ and $$[c'\in T_C]$$ are not independent.

Question 1 What is the distribution of $$\#T_C$$? Has this problem been studied before?

We have $$\#T_C=\sum_{c\in C}1_{[c\in T_C]}$$ is a sum of nonindependent Bernoulli random variables, and such sums have been studied.

Question 2 What is a good reference to learn about sums of nonindependent Bernoulli random variables ?

It seems that the random variables $$1_{[c\in T_C]}$$ and $$1_{[c'\in T_C]}$$ are positively correlated.

Question 3 Can one expect to prove something like $$\forall 0\leq k\leq 2N,\quad\mathbf{P}[\#T_C\geq k]\geq\mathbf{P}[\mathcal{B}(2N,p')\geq k]?$$

Finally, if the answer to the last question is positive, consider a generalized problem: start with independent identically distributed random subsets $$S_A\subset A$$, $$S_B\subset B$$ such that $$\forall 0\leq k\leq N,\mathbf{P}[\#S_A\geq k]\geq\mathbf{P}[\mathcal{B}(N,p)\geq k]$$, and we again have independent Bernoulli variables $$X_{a\to b},X_{b\to a}\sim\mathcal{B}(q)$$. Construct a random subset $$S_C\subset C$$ as previously: $$\begin{cases} \forall a\in A,~a\in S_C\iff a\in S_A\text{ and }\exists b\in S_B,X_{b\to a} = 1 \\ \\ \forall b\in B,~b\in S_C\iff b\in S_B\text{ and }\exists a\in S_A,X_{a\to b} = 1\end{cases}$$

Question 4 Does $$S_C$$ satisfy $$\forall 0\leq k\leq 2N,\quad\mathbf{P}[\#S_C\geq k]\geq\mathbf{P}[\mathcal{B}(2N,p')\geq k]?$$