Exponential Distribution

The exponential distribution is a very commonly used distribution in reliability engineering. Due to its simplicity, it has been widely employed even in cases to which it does not apply. The exponential distribution is used to describe units that have a constant failure rate.

The single-parameter exponential pdf is given by:

where:

λ = constant failure rate, in failures per unit of measurement, e.g. failures per hour, per cycle, etc.
λ = .
m = mean time between failures, or to a failure.
T = operating time, life or age, in hours, cycles, miles, actuations, etc.

This distribution requires the estimation of only one parameter, λ, for its application.

Statistical Properties Summary

The Mean or MTTF

The mean, , or mean time to failure (MTTF) of the 1-parameter exponential distribution is given by:

(1)

The Median

The median, , of the 1-parameter exponential distribution is given by:

(2)

The Mode

The mode, , of the 1-parameter exponential distribution is given by:

(3)

The Standard Deviation

The standard deviation, , of the 1-parameter exponential distribution is given by:

(4)

The Reliability Function

The 1-parameter exponential reliability function is given by:

This function is the complement of the exponential cumulative distribution function, or:

and:

Conditional Reliability

The conditional reliability function for the 1-parameter exponential distribution is given by:

which says that the reliability for a mission of t duration undertaken after the component or equipment has already accumulated T hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the "memoryless property."

Reliable Life

The reliable life, or the mission duration for a desired reliability goal tR for the 1-parameter exponential distribution is given by:

or:

Failure Rate Function

The exponential failure rate function is given by:

This subchapter includes the following topics: