In the Time-inhomogeneous Marriage model (Chapter 6/9.1). It sets the integral for P[Xt=M,Ct>w|Xs=B] as being from w to t-s (which given the nature of integrals is the same as from s to t-w). Are they saying then that P[Xt=M,Ct>w|Xs=B] is the same as P[Xt-w=M|Xs=B] - ie. we are only really concerned with going from B at time s to M at time t-w. But that would imply the length of time you are married doesn't come into the equaltion. Can someone shed some light on this one - thanks.
It looks to me as though you've changed the limits on the integral and then assumed it will have the same value. This won't automatically be the case, and isn't the case here as the integrand is a function of time (so integrating over a different interval will give a different answer). So, I don't think your logic holds. In particular, v in the integral shown is a measure of duration, and it appears that you've switched it to be a measure of time. Redefining your variable in this way would also affect the terms being integrated.
You're right - I managed to confuse myself there. I can see later on in the text, where we change the integral limits, we also change the function to include those changes which makes sense.
