SOA Fundamentals of Actuarial Mathematics (FAM) Practice Test

Memoryless property is what makes the exponential distribution stand out. It says that no matter how long you’ve already observed that nothing has happened, the distribution of the remaining time until the event is the same as at the start. For an exponential with rate λ, the survival function is S(x) = P(X > x) = e^{-λx}. Then the conditional probability P(X > s + t | X > s) equals S(s + t) / S(s) = e^{-λ(s + t)} / e^{-λs} = e^{-λt} = S(t). So the future is independent of the past.

This memoryless property is unique to the exponential among continuous distributions. The exponential is not symmetric around its mean; it’s skewed and defined only on [0, ∞), so symmetry doesn’t apply. It also has unbounded support, so it can take arbitrarily large values. And its hazard rate is constant (not increasing), namely λ, rather than rising with time. So the defining, distinguishing feature is its memoryless property.