Exponential Distribution's Memorylessness
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 that focuses on the timeframe within which a certain event occurs. It’s easy to assume that it can be applied in predicting lifespans or in insurance.
The Memoryless Property means that future events are not influenced by the amount of time that has already passed. For example, whether it’s a man in his 30s or 50s, if all health-related conditions are the same for both, it’s impossible to know who will die first. Even if one has lived for 20 more years, if the health conditions today are the same, the countdown to death restarts from today. More extremely, there’s no order to whether a newborn baby today or an elderly person passing away today will go first. The reason this doesn’t align with reality is that the assumption ‘all health-related conditions are the same’ is incorrect.
Conversely, if it can be demonstrated that all members of a certain group satisfy the same assumptions, their lifespans can be predicted. Insurance is exactly the process of promising a certain reward upon their death and acquiring more money in a shorter time than their expected lifespan.
Derivation
Since $P(0 \le X \le a) = 1 - e^{-\lambda a}$, then $P(X \ge a) = e^{-\lambda a}$, 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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