> [!NOTE] Definition (Normal Distribution) > Let $X$ be a [[Continuous random variables|continuous real-valued random variable]]. Let $\lambda>0.$ Then $X$ has an exponential distribution with parameter $\lambda$, denoted $X\sim \text{Exp}(\lambda),$ if its [[Probability Density Function|probability density function]] is given by $f_{X}(x)=\begin{cases}\lambda e^{-\lambda x} & \text{if } x>0 \\ 0 & x\leq 0\end{cases}$where $e^{x}$ denotes the [[Real Exponential Function|exponential function]]. # Properties By [[Exponential Distribution is Memoryless]], if $X\sim \text{Exp}(\lambda)$ then $\mathbb{P}(X > t+s \mid X>t)=\mathbb{P}(X>s).$ By [[Expectation of Exponential Distribution]], if $X\sim \text{Exp}(\lambda)$ then $\mathbb{E}[X]=\frac{1}{\lambda}.$ By [[Variance of Exponential Distribution]], if $X\sim \text{Exp}(\lambda)$ then $\text{Var}(X)=\frac{1}{\lambda^{2}}.$