Weibull Statistical Properties

The Mean or MTTF

The mean, , (also called MTTF or MTBF by some authors) of the Weibull pdf is given by:

(3)

where is the gamma function evaluated at the value of . The gamma function is defined as:

This function is provided within Weibull++ for calculating the values of Γ(n) at any value of n. This function is located in the Quick Statistical Reference of Weibull++.

For the two-parameter case, Eqn. (3) can be reduced to:

Note that some practitioners erroneously assume that η is equal to the MTBF or MTTF. This is only true for the case of β = 1 since

The Median

The median, , is given by:

(4)

The Mode

The mode, , is given by:

(5)

The Standard Deviation

The standard deviation, σT, is given by:

The Weibull Reliability Function

The equation for the three-parameter Weibull cumulative density function, cdf, is given by:

Recalling that the reliability function of a distribution is simply one minus the cdf, the reliability function for the three-parameter Weibull distribution is given by:

The Weibull Conditional Reliability Function

The three-parameter Weibull conditional reliability function is given by:

(6)

or:

Eqn. (6) gives the reliability for a new mission of t duration, having already accumulated T hours of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit or units already accumulated T hours of operation successfully.

The Weibull Reliable Life

The reliable life, TR, of a unit for a specified reliability, starting the mission at age zero, is given by:

(7)

This is the life for which the unit will be functioning successfully with a reliability of R(TR). If R(TR) = 0.50 then , the median life, or the life by which half of the units will survive.

The Weibull Failure Rate Function

The Weibull failure rate function, λ(T), is given by: