In a book about Biostatistics, I found this example to calculate expected value:
Consider the following hypothetical example of a lung cancer study in which all patients start in phase 1, transition into phase 2, and die at the end of phase 2. Unfortunately, but inevitably, all people die. Biostatistics is often concerned with studying approaches that could prolong or improve life. We assume the length of phase 1 is random and is well modeled by an exponential distribution with mean of five years. Similarly, the length of phase 2 is random and can be modeled by a Gamma distribution with parameters α = 5 and β = 4. Suppose that a new drug that can be administered at the beginning of phase 1 increases 3 times the length of phase 1 and 1.5 times the length of phase 2. Consider a person who today is healthy, is diagnosed with phase 1 lung cancer in 2 years, and is immediately administered the new treatment. We would like to calculate the expected value of the survival time for this person. Denote by X the time from entering in phase 1 to entering phase 2 and by Y the time from entering phase 2 to death without taking treatment. Thus, the total survival time is 2 + 3X + 1.5Y and the expected total survival time, in years, is E(2+3X +1.5Y) = 2+3E(X)+1.5E(Y) = 2+3×3+1.5×5/4 = 12.875 .
What I don't understand is why in the last equation E[x] is set to 3 while in my opinion it should be 5 because the length of phase 1 (for patients not taking the drug) has the exponential distribution with mean of five years?
I'm also a bit confused with this sentence:
Consider a person who today is healthy, is diagnosed with phase 1 lung cancer in 2 years, and is immediately administered the new treatment
Does that mean that the person will enter the phase 1 in the next two years while today they are healthy? It sounds kind of strange to me how medical advancements can help predict a future disease for someone's today healthy two years from now.