Which distribution is memoryless?

Which distribution is memoryless?

• A. Normal distribution
• B. Exponential distribution
• C. Poisson distribution
• D. Uniform distribution

Answer: (B)

Memoryless means the future waiting time doesn’t depend on how long you’ve already waited. The exponential distribution has this property: for any nonnegative s and t, the chance that you haven’t seen the event by time s+t given you’ve waited until time s is the same as the chance you haven’t seen it by time t. In symbols, P(X > s+t | X > s) = P(X > t). This makes the exponential distribution the natural model for waiting times in processes where events occur continuously and independently at a constant average rate (a Poisson process). Because of this, its remaining waiting time always “resets” to the same distribution, no matter how much time has already passed. Other common distributions don’t satisfy this property: for them, the conditional survival probability changes with the elapsed time, so the past affects the future. This is why the exponential distribution is the one that is memoryless.

Memoryless means the future waiting time doesn’t depend on how long you’ve already waited. The exponential distribution has this property: for any nonnegative s and t, the chance that you haven’t seen the event by time s+t given you’ve waited until time s is the same as the chance you haven’t seen it by time t. In symbols, P(X > s+t | X > s) = P(X > t). This makes the exponential distribution the natural model for waiting times in processes where events occur continuously and independently at a constant average rate (a Poisson process).

Because of this, its remaining waiting time always “resets” to the same distribution, no matter how much time has already passed. Other common distributions don’t satisfy this property: for them, the conditional survival probability changes with the elapsed time, so the past affects the future. This is why the exponential distribution is the one that is memoryless.