Exponential Statistical Properties

The Mean or MTTF

The mean, , or mean time to failure (MTTF) is given by:

Note that when γ = 0, the MTTF is the inverse of the exponential distribution's constant failure rate. This is only true for the exponential distribution. Most other distributions do not have a constant failure rate. Consequently, the inverse relationship between failure rate and MTTF does not hold for these other distributions.

The Median

The median, , is:

The Mode

The mode, , is:

The Standard Deviation

The standard deviation, σT, is:

The Exponential Reliability Function

The equation for the two-parameter exponential cumulative density function, or cdf is given by:

Recalling that the reliability function of a distribution is simply one minus the cdf, the reliability function of the two-parameter exponential distribution is given by:

One-Parameter Exponential Reliability Function

The one-parameter exponential reliability function is given by:

The Exponential Conditional Reliability

The exponential conditional reliability equation gives the reliability for a mission of t duration, having already successfully accumulated T hours of operation up to the start of this new mission. The exponential conditional reliability function is:

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.

The Exponential Reliable Life

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

or:

The Exponential Failure Rate Function

The exponential failure rate function is:

Once again, note that the constant failure rate is a characteristic of the exponential distribution, and special cases of other distributions only. Most other distributions have failure rates that are functions of time.