Exponential Distribution

def. Exponential Distribution. $X$ has exponential distribution with intensity $\lambda$:

$$\begin{array}{r}X\sim \text{Exp}(\lambda )\\ f_X(x)=\{\begin{array}{ll}\lambda e^{-\lambda x}& x>0\\ 0& \text{else}\end{array}\\ F_X(t)=1-e^{-\lambda t}\\ \mathbb{E}(X)=\frac{1}{\lambda }\mathbb{SD}(X)=\frac{1}{\lambda }\end{array}$$

• The exponential distribution is useful when modeling wait time $X$ in a call center, with “business” of $\lambda$ (e.g. “on average $\lambda$ calls per hour”).
• Be careful that $\lambda$ is the number of events in timespan, not the average time it takes between events.