Expected remaining life with constant probability of death In a famouse paper by Olivier Blanchard ("Debt, Deficits, and Finite Horizons" The Journal of Political Economy, Vol. 93, No. 2. (Apr., 1985), pp. 223-247.) he claims the following:
"If the probability of death is constant, the expected remaining life for an agent of any given age is given by $ \int_0^\infty t p e^{-pt} dt = p^{-1}$"
However, he doesn't give the justification why he used this formula.
I am trying to derive why it is so but cannot build a model that will result in such an integral. I guess he must be using some kind of discounting with continuous compounding because of the term $p e^{-pt}$ but cannot figure it out. Can anybody explain why the expected remaining life with constant probability of death $p$ results in such an integral?
P.S. Integration and the result is correct. I am interested in where did this integral come from?

First, you have to understand what it means by a constant death probability.
By constant, one may assume it is a uniform distribution. However, it does not makes sense, that a uniform distribution means the chance of dying today and dying, say, a day 1 billion years later, is the same. It is absurd for most purposes.
A more meaningful way, and in actuarial science, the usual interpretation, is that the force of mortality is constant. i.e., condition on survival, the probability you die on a short time interval divided by the time length.
The Wikipedia article is ok in providing some more context and details.
https://en.wikipedia.org/wiki/Force_of_mortality
