Exponential Distribution

The exponential distribution models the probability distribution of the time until an independent and identically distributed continuous random event occurs, with a constant hazard rate, often associated with the waiting time between Poisson-distributed events.

An exponential random variable is denoted as followed:

\[ X \sim \text{Exponential}(\beta) \]

Note that the exponential distribution is a gamma distribution that can be modeled as such:

\[ X \sim \text{Gamma}(1, \beta) \]

Probability Distribution Function (PDF)

Since the exponential distribution is modeled after the gamma distribution, we see that its pdf is:

\[\begin{split} f_X(x) = \begin{cases} \frac{1}{\beta^\alpha} e^{-\frac{x}{\beta}} , & \text{if } x>0 \\ 0, & \text{otherwise} \end{cases} \end{split}\]

Expectation

We can find the expected value of the exponential distribution from its gamma distribution origins:

\[ EX = \beta \]

Variance

We can similarly find the variance of the exponential distribution:

\[ Var(X) = \beta^2 \]