Exponential Distribution

Denoted as $\text{Exp}(\lambda)$.

$$P(x) = \lambda \exp(-\lambda x)\;\;\text{where }x \ge 0,\; \lambda \gt 0$$

Here $\lambda$ is the rate parameter, which represents the mean number of events per unit time. Similar to the rate of failures or a rate of arrivals in Poisson distribution.

Can be thought of as an continuous analogue of the geometric distribution. Often used to model the length of time until an event occurs. Memoryless.

Events must be occurring continuously and independently. Used to model inter-arrival times between completely random events (arrivals/hour), service times (services/minute), lifetime of a product which fails catastrophically (failure rate).

Properties

Relation to Poisson Distribution

If $X\sim\text{Exp}(\lambda)$ then $N \sim \text{Poisson}(\lambda)$. Here $X$ is the time until the next event, and $N$ is the number of events that occur in a fixed interval of time. The parameter is same because they describe the same underlying process.

CDF

$$F_x(x) = \lambda \int_0^x e^{-\lambda y} \text{d}y = 1 - e^{-\lambda x}$$

Mean

$$\mu_x = \frac{1}{\lambda}$$

Variance

$$\sigma_x^2 = \frac{1}{\lambda^2}$$

Percentile

$$x_{p} = \frac{-1}{\lambda} \ln \left( 1- \frac{p}{100} \right)$$

Moment Generating Function

$$M_x(t) = \frac{\lambda}{\lambda - t}$$

Theorem

$X$ has an exponential distribution iff:

• $X$ is a positive continuous r.v. and
• $X$ has memoryless property, that is $P(X>s+t | X>s) = P(X>t) \;\text{for all}\; s,t> 0$.