DES - Definition of 'Memory-Less Property'

Memory-Less Property:

The memory-less property for a probability distribution state that:

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

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~Example: where the memory-less property is NOT valid

To understand the term “memory-less” for this property, assume that $X$ models the time to failure of an electronic component.

At time (when the component is new) we compute the probability that the component does not fail.

In the next year (the first year): $P(X>s=1\text{year})$.

Then, we observe that the component does not fail in the first three years: $X>t=\text{ 3 years}$.

With these knowledge, we compute the probability that the component does not fail in the next year (the fourth year): $P(X>t+s=\text{ 4 years}\mid X>t=\text{3 years})$.

For the memory-less property, this probability equals $P(X>t=\text{1 year})$ computed when the component was new … but now is 3 years old.

$\Rightarrow$ The Memory-Less Property: the component does not “remember” that it is getting older.

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Other Useful links:

• DES - Demonstration on the Memory-Less Property of the Exponential Distribution