Hello this question is in regard to Question 4.5(iii) of the CS2 Chapter 4 end of chapter questions. The questions asks to derive a forward differential equation for the probability that a claim is yet to be logged and classified by a claims administrator at time 5, in order to obtain an expression for the probability. The model answers says this is equal to the probability that we are in state A for at least as time t. Which would be equivalent to P(W_A > t) = exp(-lambda_i * t) by use of the exponential distribution. And that is what the answer is. But what about the fact that the claim can move to state I, S or C by time t? I.e. if it moves to any other state other than L by time t. This is not equivalent to the probability that we remain in state A for at least t time though. So I guess my question is I am wondering why 1- P_AL(t) is wrong for this question.
Since we are asked for the probability that a claim is not logged by a claims administrator by time t, we only need to consider the situation where, between time 0 and t, we stay in state A (since state A is arrived in the claims dept). 1 - Pal (t) is wrong because it doesn't consider the other states we could be in at time t (I, S or C). 1 - Pal(t) - Pai (t) - Pas(t) - Pac(t) should be equivalent
