I managed a but can't do b or c.

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Best answer

Here are some hints:

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(b) and (c) can both be done as follows. Let \(t\) denote the time at which payment of a benefit commences. (And the integral w.r.t. \(t\) will be from \(0\) to \(\infty\).)

Then value, as at time \(t\), the benefit that commences then. This value will be an integral w.r.t. another variable, \(s\). The integrand you need is the same in the two cases, and involves an occupancy probability. For (b) the integral will be from \(1\) to \(6\), for (c) from \(0\) to \(\infty\).

So the result is a double integral in both cases.

There may well be other routes in both cases, but this seems simplest.

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Let me know if you need more help, but try to get it on your own!

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