What is the memoryless property of the exponential distribution?

What is the memoryless property of the exponential distribution?

• A. P(X> s+t | X> s) = P(X> s)
• B. P(X> s+t | X> s) = P(X> t)
• C. P(X> s+t | X> s) = P(X> s+t)
• D. P(X> s+t | X> s) = P(X> 0)

Answer: (B)

The memoryless property means the future waiting time does not depend on how long you’ve already waited. For an exponential distribution with rate λ, the survival function is P(X > u) = e^{-λu}. The conditional probability of waiting more than s + t given you’ve already waited past s is P(X > s + t | X > s) = P(X > s + t) / P(X > s) = e^{-λ(s + t)} / e^{-λs} = e^{-λt} = P(X > t). So the chance of surviving an additional t units is the same as if you were starting fresh, regardless of the past waiting time. This constant hazard rate means the process “forgets” how long it has already lasted. This property is characteristic of the exponential distribution (the discrete analogue is the geometric distribution).

The memoryless property means the future waiting time does not depend on how long you’ve already waited. For an exponential distribution with rate λ, the survival function is P(X > u) = e^{-λu}. The conditional probability of waiting more than s + t given you’ve already waited past s is P(X > s + t | X > s) = P(X > s + t) / P(X > s) = e^{-λ(s + t)} / e^{-λs} = e^{-λt} = P(X > t). So the chance of surviving an additional t units is the same as if you were starting fresh, regardless of the past waiting time. This constant hazard rate means the process “forgets” how long it has already lasted. This property is characteristic of the exponential distribution (the discrete analogue is the geometric distribution).