Memorylessness of the Exponential Distribution
Properties
If $X \sim \exp{ ( \lambda ) }$, then $P(X \ge s+ t ,|, X \ge s) = P(X \ge t)$
Explanation
The exponential distribution is a continuous probability distribution concerned with the duration over which a certain event occurs. Without thinking deeply, one can guess that it may be applied to lifespan prediction, insurance, and so on.
Here, the memoryless Property is the property that future events are not affected by the time that has already passed. For example, if a man in his 30s and a man in his 50s have all conditions concerning health equal, there is no way to know which of the two will die first. Even having lived 20 more years, if all conditions concerning health today are the same, the timer of death also restarts from today. More extremely, one could say there is no order to whether a newborn baby born today or tomorrow, or an elderly person on the verge of passing, goes first. The reason this does not align with reality is that the assumption that ‘all conditions concerning health are the same’ is wrong.
Conversely, if one can show that all members belonging to a certain group satisfy the same assumption, then their lifespans can be predicted. Promising a fixed compensation when their lifespans end, and collecting more money over a period shorter than the expected lifespan, is precisely insurance.
Derivation
Since $P(0 \le X \le a) = 1 - e^{-\lambda a}$, we have $P(X \ge a) = e^{-\lambda a}$, and $$ \begin{align*} P(X \ge s+ t ,|, X \ge s) =& {{P(X \ge s+ t)} \over {P(X \ge s)}} \\ =& {{e^{-\lambda (s+ t)}} \over {e^{-\lambda s}}} \\ =& e^{ - \lambda t} \\ =& P(X \ge t) \end{align*} $$
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