There is a scheduling process consisting of several hundred component tasks. The attached figure just shows a simplified diagram. The connection line with angle means that the task where the line starts should be finished before the task where the line ends.
The duration of each task follows a probability distribution. I have two questions,
1) How to compute the probability distribution of final task, e.g., task 4 in the diagram
2) If task 2b is significantly delayed, how to quantify the impact of its delay on final task, e.g., Task 4.
First, I think you are going to have to make a very charitable assumption: that the processing times of the various tasks are independent of each other. That will not be true in general. Think of the tasks being parts of a manufacturing process in a factory, and then picture a strike, power outage, fire, or other event that simultaneously affects many or all tasks. You also need to assume that processing times are independent of start or end times (workers don't lag late in the day, don't speed up when a task is overdue, ...).
So let's hold our noses and assume independence of both sorts. We'll also assume that the precedence graph is acyclic (which is highly likely). For each task $j$, let $S_j$, $P_j$ and $E_j$ be the start time, processing time and end time for that task, respectively. You are looking for the CDF of $E_4$, given the CDFs of all the $P$ variables (and the assumption that $S_j$ = 0 if task $j$ has no predecessors).
The key relationships are that $E_j = S_j + P_j$, and $S_j = \max \{E_k : (k,j) \in \mathcal{A} \}$, where $\mathcal{A}$ is the set of precedence arcs. To finish the job, you just need to know how to find the CDF of the sum of two independent variables (start time and processing time) and the CDF of the maximum of a bunch of independent variables (predecessor ending times). Both are covered in typical probability theory texts, and both can be found via Google search, so I won't reproduce them here.