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| ===Characteristics===
| | #REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Weibull_Distribution]] |
| The characteristics of the 2-parameter Weibull distribution can be exemplified by examining the two parameters, beta, <math>\beta ,</math> and eta, <math>\eta ,</math> and the effect they have on the <math>pdf,</math> reliability and failure rate functions.
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| ==== Looking at <span class="texhtml">β</span> ====
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| Beta (<span class="texhtml">β</span>) is called the shape parameter or slope of the Weibull distribution. Changing the value of <span class="texhtml">β</span> forces a change in the shape of the <span class="texhtml">''pdf''</span> as shown in the next figure. In addition, when the <span class="texhtml">''cdf''</span> is plotted on Weibull probability paper, a change in beta is a change in the slope of the distribution on Weibull probability paper.
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| <br> '''Effects of <span class="texhtml">β</span> on the ''pdf'''''
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| <br> [[Image:ALTA4.3.png|center|400px]] <br>
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| :*For <span class="texhtml">0 < β < 1</span> , the failure rate decreases with time and:
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| ::*As <math>T\to 0,</math> <math>f(T)\to \infty</math>.
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| ::*As <math>T\to \infty </math> , <math>f(T)\to 0</math>.
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| ::*<span class="texhtml">''f''(''T'')</span> decreases monotonically and is convex as <span class="texhtml">''T''</span> increases.
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| ::*The mode is non-existent.
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| :*For <span class="texhtml">β = 1,</span> it becomes the exponential distribution, as a special case, <br>
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| <br>
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| :or:
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| <br>
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| ::<math>f(T)=\frac{1}{\eta }{{e}^{-\tfrac{T}{\eta }}};\text{ }\eta >0,T\ge 0</math>
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| <br>
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| :where <math>\tfrac{1}{\eta }=\lambda =</math> chance, useful life, or failure rate.
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| <br>
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| :*For <span class="texhtml">β > 1</span>, <span class="texhtml">''f''(''T'')</span>, the Weibull distribution assumes wear-out type shapes (i.e., the failure rate increases with time) and:
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| ::*<span class="texhtml">''f''(''T'') = 0</span> at <span class="texhtml">''T'' = 0</span> .
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| ::*<span class="texhtml">''f''(''T'')</span> increases as <math>T\to \tilde{T}</math> (mode) and decreases thereafter.
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| ::*For <span class="texhtml">β = 2</span> it becomes the Rayleigh distribution as a special case. For <span class="texhtml">β < 2.6</span>, the Weibull <span class="texhtml">''pdf''</span> is positively skewed (has a right tail). For <span class="texhtml">2.6 < β < 3.7</span>, its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal <span class="texhtml">''pdf''</span>, and for <span class="texhtml">β > 3.7</span> it is negatively skewed (left tail).
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| :*The parameter <span class="texhtml">β</span> is a pure number (i.e., it is dimensionless).
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| <br> '''Effects of <span class="texhtml">β</span> on the Reliability Function and the ''cdf'''''
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| <br> [[Image:ALTA4.4.png|center|400px]]
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| <br> [[Image:ALTA4.5.png|center|400px]]
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| <br>
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| :*<span class="texhtml">''R''(''T'')</span> decreases sharply and monotonically for <span class="texhtml">0 < β < 1</span>. It is convex and decreases less sharply for the same <span class="texhtml">β</span>.
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| :*For <span class="texhtml">β = 1</span> and the same <span class="texhtml">η</span>, <span class="texhtml">''R''(''T'')</span> decreases monotonically but less sharply than for <span class="texhtml">0 < β < 1</span>, and it is convex.
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| :*For <span class="texhtml">β > 1</span>, <span class="texhtml">''R''(''T'')</span> decreases as <span class="texhtml">''T''</span> increases but less sharply than before. As wear-out sets in, it decreases sharply and goes through an inflection point.
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| <br>
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| '''Effects of <span class="texhtml">β</span> on the Failure Rate Function'''
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| <br> [[Image:ALTA4.6.png|center|400px]] <br>
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| :*The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">''T'' = 0</span>. The failure rate, <span class="texhtml">λ(''T'')</span>, decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \infty </math> or <math>\lambda (\infty )=0</math>. 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:
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| ::*Burn-in testing and/or environmental stress screening are not well implemented.
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| ::*There are problems in the production line.
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| ::*There is inadequate quality control.
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| ::*There are packaging and transit problems.
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| :*For <span class="texhtml">β = 1</span>, <span class="texhtml">λ(''T'')</span> yields a constant value of <math>\tfrac{1}{\eta }</math> , or:
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| ::<math>\lambda (T)=\lambda =\frac{1}{\eta }</math>
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| This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.
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| :*For <span class="texhtml">β > 1</span>, <span class="texhtml">λ(''T'')</span> increases as <span class="texhtml">''T''</span> increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For <span class="texhtml">1 < β < 2</span>, the <span class="texhtml">λ(''T'')</span> curve is concave. Consequently, the failure rate increases at a decreasing rate as <span class="texhtml">''T''</span> increases.
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| :*For <span class="texhtml">β = 2</span>, or for the Rayleigh distribution case, the failure rate function is given by:
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| ::<math>\lambda (T)=\frac{2}{\eta }\left( \frac{T}{\eta } \right)</math>
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| <br> Hence there emerges a straight line relationship between <span class="texhtml">λ(''T'')</span> and <span class="texhtml">''T''</span>, starting at a value of <span class="texhtml">λ(''T'') = 0</span> at <span class="texhtml">''T'' = 0</span> and increasing thereafter with a slope of <math>\tfrac{2}{{{\eta }^{2}}}</math> . Consequently, the failure rate increases at a constant rate as <span class="texhtml">''T''</span> increases. Furthermore, if <span class="texhtml">η = 1</span> the slope becomes equal to 2, and <span class="texhtml">λ(''T'')</span> becomes a straight line which passes through the origin with a slope of 2.
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| :*When <span class="texhtml">β > 2</span> the <span class="texhtml">λ(''T'')</span> curve is convex, with its slope increasing as <span class="texhtml">''T''</span> increases. Consequently, the failure rate increases at an increasing rate as <span class="texhtml">''T''</span> increases, indicating wear-out life.
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| <br>
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| ==== Looking at <span class="texhtml">η</span> ====
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| Eta (<span class="texhtml">η</span>) is called the scale parameter of the Weibull distribution. The parameter <span class="texhtml">η</span> has the same units as <span class="texhtml">''T''</span>, such as hours, miles, cycles, actuations, etc.
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| <br> [[Image:ALTA4.7.png|center|400px]] <br>
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| :*A change in the scale parameter <span class="texhtml">η</span> has the same effect on the distribution as a change of the abscissa scale.
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| ::*If <span class="texhtml">η</span> is increased while <span class="texhtml">β</span> is kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
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| ::*If <span class="texhtml">η</span> is decreased while <span class="texhtml">β</span> is kept the same, the distribution gets pushed in toward the left (i.e. toward its beginning, or 0) and its height increases.
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| <br>
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