Example: The Effect of Beta on the Weibull pdf

The following Figure shows the effect of different values of the shape parameter, β, on the shape of the $$pdf$$. One can see that the shape of the can take on a variety of forms based on the value of β.



For $$ 0<\beta \leq 1 $$:
 * As t→0 ( or γ), f(t)→∞.
 * As t→∞, f(t)→0.
 * f(t) decreases monotonically and is convex as increases beyond the value of γ.
 * The mode is non-existent.

For β &gt; 1 :


 * f(t) = 0 at ( or γ).
 * f(t) increases as $$ t\rightarrow \tilde{T} $$ (the mode) and decreases thereafter.
 * For β &lt; 2.6 the Weibull $$pdf$$ is positively skewed (has a right tail), for 2.6 &lt; β &lt; 3.7 its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal $$pdf$$, and for β &gt; 3.7 it is negatively skewed (left tail). The way the value of β relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for β = 0.999 , f(0) = ∞ , but for β = 1.001 , f(0) = 0. This abrupt shift is what complicates MLE estimation when β is close to one.