Attached is the answers to DHW exercises 9.2. b) and c). I am struggling with understanding the intuition behind these formulae. Please help :(

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The intuition behind these formulae is similar to the intuition behind the formulae associated with valuing annuities and assurances.

In both of the above cases,the integral is made up two parts - a probability of being in a certain state and a transition intensity.

For the probability - this refers to the probability that at any point over the period 0 to t of being in the state that precedes the event for which we are seeking i.e. for b) it is the probability that both lives are alive at the time t and for c) it is the probability that the life aged 40 has died and the life aged 30 is still alive at time t.

The second part of the integral is a transition intensity. This is the instaneous rate of transition from the state that precedes it (i.e. in both cases it is the rate of transition from alive to dead for the life aged 30.)

Finally, the reason why the joint life probability for the lives aged 30 and aged 40 can be separated into two probabilities is that they are independent, and hence allows for the manipulation from \(_tP_{xy}\) to \(_tP_{x}\)\(_tP_{y}\), and further allows for the expression \(_tq_{40}*_tp_{30}\).

By integrating over the period, you account for the probability of being in the state required for the event to occur at time t (in (b) it is both lives being alive, and in (c) it is only the life aged 30 being alive and the life aged 40 has died), as well as the transition intensity for the event to occur at time t (i.e. death of the life aged 30.) over every point of time during the period.