First two definition:
(A) Random Directed Graph:
Suppose we have a random $DAG(n, p)$. Here is how it's generated:
Put n distinct nodes on a line, and connect each node in the $i$th order to any node after that; This would form a complete directed graph with $n$ nodes.
Sample each edge with probability $p$.
The resulting graph will be directed, and acyclic.
(B) Transitive Closure:
Transitive closure of a graph $\text{closure}(G)$ is the result of connecting all the node pairs $i-j$ (via a direct edge $i \rightarrow j$) such that there is a path connecting $i$ to $j$.
The question is:
what is the expected number of edges in $\text{closure}(DAG(n, p))$?