Characteristics of the 2-Parameter Weibull Distribution

The characteristics of the 2-parameter Weibull distribution can be exemplified by examining the two parameters, beta, β and eta, η, and the effect they have on the pdf, reliability and failure rate functions.

Looking at β

Beta, β, is called the shape parameter or slope of the Weibull distribution. Changing the value of β forces a change in the shape of the pdf as shown in Figure 3. In addition, when the cdf is plotted on Weibull probability paper, as shown in Figure 4, a change in beta is a change in the slope of the distribution on Weibull probability paper.

Effects of β on the pdf

Fig. 3: Weibull pdf with 0 < β < 1, β = 1, β > 1 and a fixed η.

For 0 < β < 1, the failure rate decreases with time and:

As T 0, f(T) .
As T , f(T) 0.
f(T) decreases monotonically and is convex as T increases.
The mode is non-existent.

For β = 1, it becomes the exponential distribution, as a special case, or:

where = λ = chance, useful life or failure rate.

For β > 1, f(T), the Weibull assumes wear-out type shapes (i.e. the failure rate increases with time) and:

f(T) = 0 at T = 0.
f(T) increases as T (mode) and decreases thereafter.
For β = 2 it becomes the Rayleigh distribution as a special case. For β < 2.6 the Weibull pdf is positively skewed (has a right tail), for 2.6 < β < 3.7 its coefficient of skewness approaches zero (no tail); consequently, it may approximate the normal pdf and for β > 3.7 it is negatively skewed (left tail).

The parameter β is a pure number, i.e. it is dimensionless.

Effects of β on the Reliability Function and the cdf

Fig. 4: Weibull cdf, or unreliability vs. time, on Weibull probability plotting paper with 0 < β < 1, β = 1, β > 1 and a fixed η.

Fig. 5: Weibull 1-cdf, or reliability vs. time, on linear scales with 0 < β < 1, β = 1, β > 1 and a fixed η.

R(T) decreases sharply and monotonically for 0 < β < 1, it is convex and decreases less sharply for the same β.
For β = 1 and the same η, R(T) decreases monotonically but less sharply than for 0 < β < 1 and is convex.
For β > 1, R(T) decreases as T increases but less sharply than before and as wear-out sets in, it decreases sharply and goes through an inflection point.

Effects of β on the Failure Rate Function

Fig. 6: Weibull failure rate vs. time with 0 < β < 1, β = 1, β > 1

The Weibull failure rate for 0 < β < 1 is unbounded at T = 0. The failure rate, λ(T), decreases thereafter monotonically and is convex, approaching the value of zero as T or λ() = 0. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When such behavior is encountered, one or more of the following conclusions can be drawn:

Burn-in testing and/or environmental stress screening are not well implemented.
There are problems in the production line.
Inadequate quality control.
Packaging and transit problems.
For β = 1, λ(T) yields a constant value of , or:

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For β > 1, λ(T) increases as T increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For 1< β < 2 the λ(T) curve is concave, consequently the failure rate increases at a decreasing rate as T increases.
For β = 2, or for the Rayleigh distribution case, the failure rate function is given by:

hence there emerges a straight line relationship between λ(T) and T, starting at a value of λ(T) = 0 at T = 0 and increasing thereafter with a slope of . Consequently, the failure rate increases at a constant rate as T increases. Furthermore, if η =1 the slope becomes equal to 2 and λ(T) becomes a straight line which passes through the origin with a slope of 2.

When β > 2 the λ(T) curve is convex, with its slope increasing as T increases. Consequently, the failure rate increases at an increasing rate as T increases indicating wear-out life.

Looking at η

Eta, η, is called the scale parameter of the Weibull distribution. The parameter η has the same units as T, such as hours, miles, cycles, actuations, etc.

Fig. 7: Weibull pdf with η = 50, η = 100, η = 200

A change in the scale parameter η has the same effect on the distribution as a change of the abscissa scale.

If η is increased while β is kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
If η is decreased while β is kept the same, the distribution gets pushed in toward the left (i.e. toward its beginning, or 0) and its height increases.