Memoryless Property

A random variable $X$ is memoryless if:

$P(X>s+t\mid X>t)=P(X>s)$

for all $s,t\ge 0$.

Intuition

If the lifetime of an item is memoryless, an item that has been in use for $t$ hours is as good as a new item regarding the amount of time remaining until it fails. The item does not “remember” how long it has already been in use.

Uniqueness

The exponential distribution is the only continuous distribution with this property. This is why it appears so frequently in stochastic process modeling — it’s the natural choice when “the future doesn’t depend on the past.”

Everyday Illustration

• Memoryless: A lightbulb with exponential lifetime. A bulb that has lasted 1000 hours has the same expected remaining lifetime as a brand new bulb.
• Not memoryless: A car tire. A tire that has been driven 50,000 km is more likely to fail soon than a new tire — it “remembers” its wear.

Connection to Stochastic Processes

The memoryless property of the exponential distribution is what makes the Poisson Process “restart” probabilistically at each event. This is why inter-arrival times are i.i.d. exponential — each event resets the clock.

Related

• Exponential Distribution
• Inter-arrival Times
• Failure Rate Function