[[Exponential distribution]]
# Memoryless distribution

A [[real random variable]] $X : \xi \to \mathbb{R}$ is said to be **memoryless** iff
$$
\begin{align*}
\mathbb{P}(X \geq s + t \mid X \geq s) = \mathbb{P}(X \geq t)
\end{align*}
$$
or equivalently
$$
\begin{align*}
\mathbb{P}(X\geq s)\mathbb{P}(X\geq t)=\mathbb{P}(X \geq s+t)
\end{align*}
$$
for all $s,t > 0$.
A positive [[continuous random variable]] is memoryless iff it has the [[exponential distribution]] #m/thm/prob 

> [!missing]- Proof
> #missing/proof

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#state/tidy | #lang/en | #SemBr