Questions: Exponential Distribution: Waiting Times and Lifetimes

Question 1

A machine component has an exponentially distributed lifetime with mean 100 hours. You check on it at hour 80 and it is still working. How does its remaining expected lifetime compare to that of a brand-new component?

• A. It is shorter — the component has aged and is more likely to fail soon
• B. It is longer — a component that survived 80 hours must be more reliable than average
• C. It is exactly the same — 100 hours remaining, as if it were brand new
• D. It cannot be determined without knowing the specific failure mechanism

Answer: (C)

This is the memoryless property: P(X > 80 + t | X > 80) = P(X > t). A component that has survived to age 80 has exactly the same remaining lifetime distribution as a brand-new component — 100 hours expected remaining. This seems counterintuitive because for most physical systems, age implies wear. But the exponential distribution is specifically characterized by the absence of aging: past survival gives zero information about future failure. Options A and B both import reasoning about wear-out or selection that applies to non-exponential distributions.

Question 2

Events occur as a Poisson process with rate λ = 3 events per hour. What is the probability that you wait more than 30 minutes (0.5 hours) for the next event?

• A. P(X > 0.5) = e^{−3} ≈ 0.050
• B. P(X > 0.5) = e^{−1.5} ≈ 0.223
• C. P(X > 0.5) = 1 − e^{−1.5} ≈ 0.777
• D. P(X > 0.5) = e^{−0.5} ≈ 0.607

Answer: (B)

Inter-arrival times for a Poisson process with rate λ = 3 are Exp(λ = 3). The survival function is P(X > t) = e^{−λt} = e^{−3(0.5)} = e^{−1.5} ≈ 0.223. Option A uses t = 1 instead of t = 0.5. Option C is the CDF (probability of waiting less than 0.5 hours), not the survival function. Option D uses λ = 1 instead of λ = 3. The connection between the Poisson rate and the exponential rate parameter is the key: λ appears in both distributions and links counts to waiting times.

Question 3

A light bulb with exponentially distributed lifetime has been burning for 1,000 hours without failing. It is now more likely to burn out in the next hour than it was when it was new.

• T. True
• F. False

Answer: False

This directly contradicts the memoryless property. For an exponential distribution, the hazard rate (probability of failure in the next instant given survival so far) is constant at λ — it never increases with age. The probability of burning out in the next hour is exactly the same whether the bulb is new or has been running for 1,000 hours. This is what distinguishes the exponential from distributions like the Weibull (with increasing hazard rate, modeling wear-out) or the Weibull with decreasing hazard rate (infant mortality).

Question 4

The exponential distribution is the only continuous probability distribution with the memoryless property.

• T. True
• F. False

Answer: True

This is a theorem, not just a claim. The memoryless property P(X > s + t | X > s) = P(X > t) imposes the functional equation S(s + t) = S(s) · S(t) on the survival function. Among continuous functions with S(0) = 1 and S decreasing, the only solutions are S(x) = e^{−λx} for λ > 0 — precisely the exponential distributions. The geometric distribution is the discrete analogue (the only discrete memoryless distribution). Any other continuous distribution you might consider — normal, gamma, Weibull — fails the memoryless property.

Question 5

Explain in your own words what the memoryless property means and why it is surprising compared to how most real-world lifetimes behave.

Model answer: The memoryless property means that a component's remaining lifetime distribution is identical regardless of how long it has already been running. Knowing it has survived to time s gives no information about how much longer it will last — the remaining lifetime looks just like starting fresh. This is surprising because most real objects show aging: a used car, a biological organism, or a mechanical component typically has a higher failure rate the older it gets. The exponential distribution represents a theoretically idealized system with no aging — a constant hazard rate throughout its life. In practice, this is approximately true for random failures (cosmic ray hits, random power surges) but not for wear-out failures.

The memoryless property is the defining characteristic of the exponential distribution, and understanding why it is unusual helps clarify when the exponential is and is not an appropriate model. It applies well to radioactive decay, random hardware failures due to external shocks, and interarrival times in Poisson processes — all settings where there is no accumulating degradation. It fails for biological aging, fatigue, and wear, which are better modeled by distributions with increasing hazard rates.