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?
 A: The distribution of positive failure time with constant "instantaneous event rate" (=constant hazard rate) is the exponential distribution. It has the "memoryless property" that you mention.
You can see this e.g. by considering the conditional expectation of an exponential random variable $X \sim \text{Exp}(\lambda)$ conditional on $X>t$ for $\lambda>0$ and $t>0$. The probability density function of $X$ is $f(x) = \lambda e^{-\lambda x}$ and thus, you get:
$$E(X | X>t) - t = \frac{1}{P(X>t)} \times \int_t^\infty x \times \lambda e^{-\lambda x} dx - t \\
= e^{\lambda t} \times e^{-\lambda t} (t + 1/\lambda) - t \\
= 1/\lambda$$
If you want to build models that make the assumption of a constant reference hazard rate, there is plenty of software that implements exponential time to even regression. You can also use software for Poisson regression by providing either the event count as 1 (=event) or 0 (=censored), and the natural logarithm of the time to first event or censoring as an offset.
A: 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
