
Actuary and probablity
The probability of death for a person within the next one year is 1%. His death rate is unrealistically timeinvariant, like a robot.
What is this person's life expectancy at the time of his birth? What is the probability that he survives up to this life expectancy?
If he lives for 65 years, what is his remaining life expectancy?

Assuming I read your question right: Every year he has probability p=0.01 of dying and this is fixed throughout his life such that every year he has probability (1p)=0.99 of surviving that year and each year is independent of the last.
No guarantees that this is correct but I would think of it this way. The person's life in years can be thought of as a Geometric Random Variable with parameter p=0.01 so you want to ask how many years until death:
Assuming you consider the year in which they die as a part of their life.
The expectation of a geometric random variable isn't that hard to deduce (or look up).
The other thing to know is that, like the exponential distribution, the Geometric distribution has the memoryless property which should answer your last question.
This is of course pretty ridiculous because the probability of you dying is certainly dependent on how many years you have lived.