# Exponential



Exponential Distribution has the following properties:

1. Equals the distance between successive occurances or arrivals of a Poisson process with mean $\lambda > 0$
2. $\lambda$ is the average number of occurances or arrivals per unit of time (length, space, etc.)
3. $\frac{1}{\lambda}$ is the average time between occurrences or arrivals.

\defl{Exponential Distribution:}
$f(x) = \lambda e^{-{\lambda}x}$
$F(x) = 1- e^{-{\lambda}x}$

• $0 \leq x < \infty$
• $\mu = E(X) =\frac{1}{\lambda}$
• $\sigma^2 = V(X) = \frac{1}{\lambda^2}$
\defs{Examples}
• The amount of time until the next customer at The Pizza Customer will arrive.
• The amount of time until the DVD player will break. The exponential distribution is very useful for determining length of warranty.
• The amount of time until the next person will arrive at a specific ATM