Why Is the Weibull Distribution Important in Mathematics?
Weibull Distribution
The Weibull distribution is a continuous probability distribution. It is one of the most used lifetime distributions that has applications in reliability engineering. It is an adaptable distribution that can take on the features of other kinds of distributions, depending on the value of the shape parameter. It is used to analyse the life data and helps to access the reliability of the products. In this article, we would discuss what is the Weibull distribution, what is the Weibull distribution formula, the properties, reliability, Weibull distribution examples, two-parameter Weibull distribution, and inverse Weibull distribution in depth for your better understanding.
Weibull Distribution Definition
Weibull distribution is a type of continuous probability distribution that is used in analysing life data, times of model failure, and for accessing product reliability. It can also fit in a wide range of data from several other fields like hydrology, economics, biology, and many engineering sciences. It makes for an extreme value of probability distribution that is often used to model reliability, wind speeds, survival, and several other data. The main reason for using Weibull distribution is due to its flexibility since it can simulate several other distributions just like exponential and normal distributions. Weibull distribution reliability can be measured with the help of two parameters. Two different Weibull probability density function, also called as Weibull distribution pdf are commonly used: two-parameter pdf and three-parameter pdf.
Weibull Distribution Formula
Let us now take a look at the Weibull formula.
The general expression of the Weibull pdf is noted by the three-parameter Weibull distribution expression which is given by:
f(T) = \[\frac{\beta}{\eta}\] \[(\frac{T-\gamma}{\eta})^{\beta-1}\] \[e^{(\frac{T-\gamma}{\eta})\beta}\]
wherein,
f(T) \[\geq\] 0 T \[\geq\] 0 or \[\gamma\],\[\beta\]> 0, \[\eta\]> 0, - \[\infty\] < \[\gamma\]< \[\infty\]
and:
β is called the shape parameter, also called as the Weibull slope
η is called the scale parameter
γ is called the location parameter
Usually, the location parameter is not much used, and you can set the value of this parameter to zero. When this is done, the pdf equation reduces to the two-parameter Weibull distribution.
Two-Parameter Weibull Distribution
The formula of the two-parameter Weibull distribution is practically much similar to the three-parameter Weibull distribution, the only difference being that μ isn’t included:
The two-parameter Weibull is commonly used in failure analysis since no failure happens before time zero. If you know μ, the time when this failure happens, you can easily subtract it from x (i.e. time t). Hence, when you shift from the two-parameter to the three-parameter distribution, all you need to do is simply replace every instance of x with (x – μ).
Weibull Distribution Reliability
The Weibull distribution is commonly used in the analysis of reliability and life data since it could adapt to different situations. Depending upon the parameter values, this distribution is used for modelling a variety of behaviours for a specific function. The probability density function generally describes the distribution function. The parameters of the distribution control the location, scale, shape, of the probability density function. Many methods are used for measuring the reliability of the data. However, the Weibull distribution method is amongst the best methods for analysing the life data.
Properties of Weibull Distribution
The properties of Weibull distribution are as follows:
Cumulative distribution function
Probability density function
Shannon entropy
Moments
Moment generating function
Inverse Weibull Distribution
The inverse Weibull distribution could model failure rates that are much common and have applications in reliability and biological studies. A three-parameter generalized inverse Weibull distribution that has a decreasing and unimodal failure rate is presented and studied. Similar to the Weibull distribution, the three-parameter inverse Weibull distribution is presented for studying the different density shapes and functions of the failure rate.
The probability density function of the inverse Weibull distribution is as follows:
f(x)=γαγx−(γ+1) exp[−(αx)γ]
Weibull Distribution Example
The Weibull distribution is commonly used in the analysis of reliability and life data since it is much versatile. Depending on the parameter values, the Weibull distribution is used to model several life behaviours.
Weibull Distribution Solved Examples
1. Calculate the Weibull distribution whose α & β is 2 & 5, X1 = 1, X2 = 2.
Solution:
The first step is to substitute all these values in the above formulas.
P(X1 < X < X2) = e-(X1/β)α - e-(X2/β)α
P(1 < X < 2) = e-(1/5)2 - e-(2/5)2
= 0.9608 - 0.8521
= 0.1087
Then calculate the mean:
Use the formula μ = βΓ(1 + 1/α)
= 5x Γ(1+1/2)
= 5x Γ(1.5)
= 5x 0.8864
= 4.432
The next step is to calculate the median:
Use the formula β(LN(2))1/α
= 5x(0.6932)(1/2)
= 5x0.8326
= 4.1629
Next, calculate the variance:
Use the formula σ2 = β2 [Γ(1 + 2/α) - Γ(1 + 1/α)2]
σ2 = 52 [Γ(1 + 2/2) - Γ(1 + 1/2)2]
= 25x[Γ(2)- Γ(1.5)2]
= 25x[1- 0.7857]
= 25X 0.2143
= 5.3575
Lastly, calculate the standard deviation:
σ = √value of variance
= √(5.3575)
= 2.3146
Facts About Weibull Distribution
1. The Weibull distribution can assume the characteristics of several different types of distributions. For this reason, it is extremely popular amongst the engineers and quality practitioners, who made it the commonly used distribution to model reliability data.
2. Its flexibility is the reason why engineers use the Weibull distribution for evaluating the reliability and material strengths of almost every type of things ranging from capacitors and vacuum tubes to relays and ball bearings.
3. The Weibull distribution also can model hazard functions that are increasing, decreasing, or constant, and allows it to describe any kind of phase of any item’s life.
FAQs on Weibull Distribution: Meaning, Formula & Examples
1. What is the Weibull distribution and where is it commonly used?
The Weibull distribution is a continuous probability distribution that is highly versatile for modelling various phenomena. It is most frequently used in reliability engineering and survival analysis to model the time-to-failure of components or systems. Its applications also extend to weather forecasting (modelling wind speeds), insurance (modelling claim sizes), and quality control.
2. What are the main parameters of a 2-parameter Weibull distribution and what do they represent?
A 2-parameter Weibull distribution is defined by its shape and scale parameters, which give it great flexibility:
- Shape Parameter (β or k): This is the most crucial parameter. It determines the shape of the distribution curve and describes the nature of the failure rate over time.
- Scale Parameter (η or λ): This parameter, also known as the characteristic life, stretches or compresses the distribution along the time axis. It represents the time at which 63.2% of the population is expected to have failed.
3. How does the value of the shape parameter (β) affect the failure rate in a Weibull analysis?
The shape parameter (β) directly indicates how the failure rate of an item changes over its lifetime. This is a key reason for its popularity in reliability studies.
- β < 1: Indicates a decreasing failure rate. This is common in early life failures or "infant mortality" where defective products fail early.
- β = 1: Indicates a constant failure rate. This means the distribution is identical to the exponential distribution, suggesting random failures that are independent of age.
- β > 1: Indicates an increasing failure rate. This represents wear-out failures, where the probability of failure increases as the product ages.
4. What is the primary difference between a Weibull distribution and a Normal distribution?
The main difference lies in their shape and application. The Normal distribution is always symmetric (a perfect bell curve) and is used for data that clusters around a central mean. In contrast, the Weibull distribution is typically skewed and is bounded at zero, making it much more flexible. This makes it ideal for modelling real-world phenomena like lifetime or time-to-failure, which cannot be negative and often don't fail symmetrically.
5. What is the formula for the Probability Density Function (PDF) of a Weibull distribution?
The Probability Density Function (PDF) for a 2-parameter Weibull distribution is given by the formula:
f(t; β, η) = (β/η) * (t/η)β-1 * e-(t/η)β
Here, t represents time, β is the shape parameter, and η is the scale parameter (characteristic life). This formula calculates the probability of failure at a specific time t.
6. Why is the Weibull distribution often preferred over the exponential distribution for reliability analysis?
The exponential distribution is a special case of the Weibull distribution where the shape parameter β = 1. This means it can only model a constant failure rate. The Weibull distribution is more versatile because its shape parameter can also model decreasing (β < 1) and increasing (β > 1) failure rates. This allows it to accurately represent all three phases of a real-world product life cycle: infant mortality, useful life, and wear-out, which the exponential distribution cannot do.
7. How can you estimate the parameters of a Weibull distribution from a set of data?
There are two primary methods for estimating the Weibull parameters (shape and scale) from failure data. The first is Weibull probability plotting, a graphical method where data points are plotted on special paper to form a straight line, from which the parameters can be visually estimated. The second, more common method is Maximum Likelihood Estimation (MLE), a statistical technique that uses numerical methods to find the parameter values that best fit the observed data.
8. Can a Weibull distribution only model time-to-failure data?
No, while its most famous application is in reliability and survival analysis, the Weibull distribution's flexibility allows it to model a wide range of continuous, non-negative data. For instance, it is effectively used in meteorology to model wind speed distributions, in finance to model stock returns, and in manufacturing to model the size of particles.
