Consider two Poisson Random Processes $A(t), B(t)$ with parameters $a, > b$. Let $T$ be the time where exactly two events of type A and exactly two events of type B have occurred. What distribution does $T$ follow?
My attempt:
Consider $X = X_1 + X_2$ the time that two events of type A have occurred and $Y = Y_1 + Y_2$ the time when two events of type B have occurred. The interarrival times follow exponential distribution so $X \sim G(2, a), Y \sim G(2, b)$ Let $T = \max (X, Y)$
Then $$F_T(t) = \Pr[T \le t] = \Pr [X \le t] \Pr [Y \le t] = F_X(t)F_Y(t) $$ and its PDF is $$f_T(t) = f_X(t) F_Y(t) + f_Y(t) F_X(t)$$
Is my approach right?
Yes, but to justify it, you need to use that $\{A(t)\ge 2\}=\{X\le t\}$ and $\{B(t)\ge 2\}=\{Y\le t\}$. In particular, $T= \min\{t\ge 0: A(t)\ge 2, B(t)\ge 2\} = \min\{t\ge 0: X\le t, Y\le t \}=\max(X,Y) $. Then if both processes are independent, you can use your argument.