## The Exponential Distribution

The exponential distribution is illustrated in Figure 11.3 and has the following density function:

Also shown in the figure is the CDF, which starts at zero and approaches the value of unity asymptotically.

We have seen the exponential distribution as derivable from the Poisson when k = 0. This distribution is widely used in reliability theory wherein the value of X is taken to be a constant failure rate for a part of a system (e.g., a component). In such a case, variable x is converted into a time variable, t, and the CDF is found by integrating the preceding relationship, yielding

Because this is a failure distribution (i.e., represents the failure behavior), it can be converted into a reliability formula as (a) The density function x (a) The density function

(b) The cumulative distribution function

Figure 11.3. The exponential distribution.

where R(t) is the probability of successful (failure-free) operation to time t. This is the very familiar expression of the reliability of a system, or component, with a constant failure rate. As indicated in Section 8.7.2 of Chapter 8, the failure rate and the mean time between failures (MTBF) are reciprocals of one another. The MTBF, or 1/X, is the mean value of this distribution because it represents the mean time to failure. 