The solutions go through the way the EPV is calculated. I'm not sure from what you've written where your problem lies.... If it's to do with the inital integral expression, to get this you need to think about the different circumstances in which a benefit would be paid. To start off, let's assume the the insured life dies within the first 15 years. 15 years is a sensible point to split things down as if we're within 15 years the second condition simplifes to "his death occurs after the death of another life now aged exactly 45". So for a payment to be made within the first 15 years: - we need 10 years to have passed as the insured life must be over 50. So we're down to thinking about him dying in 10 to 15 years from now. Hence the integral limits. - we need the insured life to die at time t, giving the p x mu factor - we need the 45 year old to be already dead, giving a q factor (re-written as a 1-p). - then we need to add in discounting (to allow for interest over the intervening period) and integrate over our possible time values. Putting this together gives the first part of the solution. The second part can be reasoned similarly, although then you're thinking about what happens if the insured life dies more than 15 years in the future. Hope all this helps.
It's the probability of the 45 year old dying in the 15 years before t (when we're assuming the 40 year old dies). This is necessary to satisfy condition 2. i.e. we need the 45 year old to be alive at time t-15, but dead by t. i.e. we need the 45 year old to survive t-15 years, but not survive more than t years. i.e. we need a factor of (t-15)p45 - (t)p45 If you're not happy with the logic from the second to third step above, you can do it algebraically starting from (t-15)p45 x (15)q45+t-15.
