> [!NOTE] Theorem > The [[Exponential Distribution|exponential distribution]] is a [[Memoryless Probability Distribution|memoryless distribution]]: that is, if $X\sim \text{Exp}(\lambda)$ then for all $s,t \geq 0$ $\mathbb{P}(X>t+s\mid X>t)=\mathbb{P}(X>s)$where $\mathbb{P}(A\mid B)$ denotes the [[Conditional Probability|conditional probability]] of $A$ given $B.$ **Proof**: Let $s,t\geq 0.$ Then $\mathbb{P}(X>t)=e^{-\lambda t}.$ Thus $\mathbb{P}(X>t+s\mid X>t)=\frac{\mathbb{P}(X>t+s)}{\mathbb{P}(X>t)}=\frac{e^{-\lambda(t+s)}}{e^{-\lambda t}}=e^{-\lambda s}= \mathbb{P}(X>s).$