Weibull Distribution Weibull Distribution In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fr茅chet (1927) and first applied by Rosin & Rammler (1933) to describe the size distribution of particles. The probability density function of a Weibull random variable x is: where k > 0 is the shape parameter and 位 > 0 is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential function. The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the exponential distribution (k = 1) and the Rayleigh distribution (k = 2). If the quantity x is a "time-to-failure", the Weibull distribution gives a distribution for which the failure rate is proportional to a power of time. The shape parameter, k, is that power plus one, and so this parameter can be interpreted directly as follows:
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A value of k=1 indicates that the failure rate is constant over time. This might suggest random external events are causing mortality, or failure. A value of k>1 indicates that the failure rate increases with time. This happens if there is an "aging" process, or parts that are more likely to fail as time goes on. In the field of materials science, the shape parameter k of a distribution of strengths is known as the Weibull modulus. Example :Calculate the probability that a part will fail at time t = 2 if that parts failure occurrence is Weibull distribution and has, the shape parameter, γ = 0.5 and, scale parameter, α = 4. Solution:The problems says calculate the probability that a part will fail at time t = 2 So let x = 2 γ = 0.5 Here 0.5 < 1 so failure rate decreases over time. α=4 Now here we will have two turning points. Whether they are asking for cumulative distribution function or Probability density function? The graph will vary according to that. Here as the problem wants the answer at exactly at t = 2, we use Probability density function. So when the table as given is checked at t = 2 Any change in the scale parameter α, will have same effect on the probability distribution as that of the change of the x axis scale.
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The probability density function stretches out as the scale parameter α increases with γ being kept constant. Lets assume the other two parameters γ shape parameter, and μ location parameter are kept constant, then we can see changes in the α, scale parameter. If α is increasing, the shape and location will remain same but the distribution will get stretched out to the right as well as the height decreases. If α is decreased, the shape and location will still remain same but the distribution will get pushed in towards left as well as the height increase. The unit of α will be same as the unit taken for x.
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