Exponential

Parameters: $\lambda$

The exponential distribution models the amount of time until an event occurs.

Note that this is the continuous equivalent of the geometric distribution.

The probability density function is

$$f(x) = \lambda e^{-\lambda x} \; \; \; \mbox{ for } x \ge 0$$ (7.10)

Note that the exponential distribution is the same as the gamma distribution with parameters $(1, \lambda)$. (I.e. time until first event, $n=1$.)

The mean and variance of $x$:

$$\begin{eqnarray*} \mu &=& \frac{1}{\lambda}\\ \sigma^2 &=& \frac{1}{\lambda^2} \end{eqnarray*}$$

The cumulative distribution function is

$$F(x) = 1 - e^{-\lambda x}$$ (7.11)

The exponential distribution is memoryless:

$$P(X > t+s \; \mid \; X > t) = P(X > s) \; \; \mbox{for all } s, t \ge 0 ,$$ (7.12)

meaning that if the instrument is alive at time $t$, the probability of survival to time $t+s$ (i.e., from time $t$ to $s$) is the same as the initial probability of surviving to time $s$. (I.e., the instrument doesn't remember that it already survived to $t$).