View source for Template:The effect of beta on Weibull failure rate

'''The Effect of <span class="texhtml">β</span> on the Weibull Failure Rate'''

The value of <span class="texhtml">β</span> has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of <span class="texhtml">β</span> is less than, equal to, or greater than one. 


[[Image:WB.8 weibull failure.png|center|250px| The effect of <math>\beta</math> on the Weibull failure rate function. ]]


As indicated by above Figure, populations with <span class="texhtml">β &lt; 1</span> exhibit a failure rate that decreases with time, populations with <span class="texhtml">β = 1</span> have a constant failure rate (consistent with the exponential distribution) and populations with <span class="texhtml">β &gt; 1</span> have a failure rate that increases with time.  All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of <span class="texhtml">β.</span> The Weibull failure rate for <span class="texhtml">0 &lt; β &lt; 1</span> is unbounded at  <span class="texhtml">(</span>or <span class="texhtml">γ)</span>. The failure rate, <span class="texhtml">λ(''t''),</span> decreases thereafter monotonically and is convex, approaching the value of zero as <span class="texhtml">''t''→∞</span> or <span class="texhtml">λ(∞) = 0</span>. 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 encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For <span class="texhtml">β = 1</span>, <span class="texhtml">λ(''t'')</span> yields a constant value of <math> { \frac{1}{\eta }} </math> or: 

::<math> \lambda (t)=\lambda ={\frac{1}{\eta }} </math> 

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

For <span class="texhtml">β &gt; 1</span>, <span class="texhtml">λ(''t'')</span> increases as  increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For <span class="texhtml">1 &lt; β &lt; 2,</span> the <span class="texhtml">λ(''t'')</span> curve is concave, consequently the failure rate increases at a decreasing rate as  increases. 

For <span class="texhtml">β = 2</span> there emerges a straight line relationship between <span class="texhtml">λ(''t'')</span> and , starting at a value of <span class="texhtml">λ(''t'') = 0</span> at <span class="texhtml">''t'' = γ</span>, and increasing thereafter with a slope of <math> { \frac{2}{\eta ^{2}}} </math>. Consequently, the failure rate increases at a constant rate as  increases. Furthermore, if <span class="texhtml">η = 1</span> the slope becomes equal to 2, and when <span class="texhtml">γ = 0</span>, <span class="texhtml">λ(''t'')</span> becomes a straight line which passes through the origin with a slope of 2. Note that at <span class="texhtml">β = 2</span>, the Weibull distribution equations reduce to that of the Rayleigh distribution. 

When <span class="texhtml">β &gt; 2,</span> the <span class="texhtml">λ(''t'')</span> curve is convex, with its slope increasing as  increases. Consequently, the failure rate increases at an increasing rate as  increases indicating wear-out life. 

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