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Weibull Distribution is a continuous probability distribution that is very important in reliability engineering. With the help of shape parameters, it can take the values from the other distributions. It is a two-parameter family of the curve which serves as a perfect analytical tool for modeling the breaking strength of materials. The study of Warranty analysis, utility services, and parts like bearing, capacitors, etc manufactured in the factory are examples of Weibull distribution. In Weibull's distribution, an item’s constancy is analyzed and the item’s failure is determined by data analysis.
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Key Terms: Weibull distribution, reliability, two-parameter
Weibull Distribution Formulas
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There are two types of Weibull probability density functions (pdfs). They are
- Two parameter pdf
- Three parameter pdf
The formula for two-parameter pdf is
F(x) = Ƴ ((x))Ƴ-1 exp(-((x))Ƴ) x≥0
The value of the shape parameter Ƴ determines the failure rate
- The failure rate decreases with time when y<1
- The failure rate is constant when y=1
- The failure rate increases with time when y>1
The formula for a three-parameter pdf is:
F(x) = Ƴ ((x-µ))Ƴ-1 exp(-((x))Ƴ) x≥µ; Ƴ,α > 0
Or
F (x) = ƳxƳ-1 exp(-x)Ƴ , x ≥ 0; y> 0
- Ƴ = shape parameter, Weibull slope, threshold parameter
- α= scale parameter, characteristic life parameter
- µ = location parameter, waiting for time parameter, the shift parameter
Weibull Distribution
Weibull Distribution Reliability
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Weibull distribution has a unique characteristic to adapt to different situations which is why it is primarily used in reliability and life data analysis. Parameter values affect the distribution which is used to model the different behaviors for a particular function.
Distribution function is generally determined by the probability density function. The scale, shape, and location of the probability density function are managed by the parameters in the distribution. Weibull distribution is considered to be one of the best methods to calculate life data among several other methods used to calculate the reliability of data.
Plot of Weibull Distribution
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Plot of Weibull Distribution is a graphical representation that determines whether a data set can be created from a population that can unavoidably satisfy the two-parameter Weibull distribution where the location is anticipated to be zero. The design of the plot is unique as it determines whether the data supports Weibull distribution and if it does then the points will be linear or approximately linear.
Properties of Weibull distribution
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Some of the properties of Weibull distribution are:
- Shannon entropy
- Moment generating function
- Probability density function
- Moments
- Cumulative distribution function
Inverse Weibull Distribution
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Reliability and biological study areas take inverse Weibull distribution into consideration as it can model failure rates. The inverse Weibull distribution is a three-parameter probability density function that is used to study density shapes and failure rate function. the formula of inverse Weibull distribution is:
F(x) = ƳαƳx-(Ƴ+1)exp [-()Ƴ]
Things to Remember
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- Weibull’s distribution reliability is measured by the parameters
- Two parameters and three-parameter are the two types of Weibull probability density function
- The time failure of an item being always positive makes the Weibull distribution a continuous function.
- Weibull distribution can acquire the characteristics of different types of distributions.
- It is used in model reliability data by engineers and quality practitioners.
- Because of its flexibility, the Weibull distribution is used for analyzing the reliability and material strength
- Statistical software programs are used to perform the Weibull analysis process.
Sample Questions
Ques. A magnetic disk is exposed to corrosive gas. What will be the probability of it failing before 500 hours given the value of Ƴ=300 and α=0.5? (3 Marks)
Ans: F(x) = 1- exp^(-(x/α)Ƴ)
The probability of the failure of the disk before 500hours = P(x0.5)
=1- exp^(-(1.6667)0.5)
=0.725
Ques. A magnetic disk is exposed to corrosive gas. What will be the probability of it to last 600 hours or more? (3 Marks)
Ans: F(600) = 1-exp^(-(600/300)0.5)
P (x>= 600) = 1-P (x1 – exp ^(-(600/300)0.5)
=0.2431
Ques. The time failure of a bearing’s Weibull distribution with the parameters Ƴ = 0.5, α= 5000, and µ= 0. Calculate the probability of a bearing lasting for 6000 hours. (2 Marks)
Ans: 1 - exp^((6000/5000)0.5)
= 0.666
Ques. What is the process of fitting a Weibull distribution with the help of regression? (4 Marks)
Ans: there is a way to calculate the parameters for a Weibull distribution with the help of linear regression. The carminative distribution function of Weibull distribution is expressed in the linear equation as:
Y = ln(-ln(1-F(x))), x’ = ln x and a = -ß ln α
It is possible to find a coefficient with the help of linear regression if the sample has a Weibull distribution.
Ques. What is the method of increasing the height of the Weibull distribution graph? (2 Marks)
Ans: the height is increased as the distribution is pushed to its left by keeping ß and y constant and decreasing Æ. The consistency of the material and the narrow probability curve of the strength distribution is detected by the high graph of the Weibull distribution.
Ques. The time failure of a gadget follows Weibull's distribution. If the scale = 1000 hours and shape = 0.5, what is the meantime to failure? (3 Marks)
Ans: when the failure of the gadget follows a Weibull distribution, the meantime to failure :
1000 x Ð (1 + 1/0.5)
= 1000 x 2
= 2000 hours
Ques. Where will be the inflection point for the Weibull function if ß is greater than 1? (2 Marks)
Ans: (e1/ß- 1)/e1/ß is the inflection point for the Weibull function. The function changes from convex to concave after the inflection point. ß is the shape parameter of the Weibull distribution. It represents the failure rate behavior. The failure rate increases with time if ß is greater than 1 . if the failure rate decreases with time, the ß is less than 1. The failure rate is static if ß is equivalent to 1.
Ques. How does the threshold parameter y change in the graph of Weibull distribution? (2 Marks)
Ans: ‘y’ is known as the location parameter as it determines the minimum time of failure of the device. The graph shifts towards the right as y increases.
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