Weibull Distribution formula properties and solved examples
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 in Probability and Statistics
1. What is the Weibull distribution?
The Weibull distribution is a continuous probability distribution used to model time-to-failure and reliability data. It is defined by two main parameters:
- Shape parameter (k) – controls the failure rate behavior.
- Scale parameter (λ) – stretches or compresses the distribution.
2. What is the formula for the Weibull probability density function (PDF)?
The Weibull PDF is given by f(x) = (k/λ)(x/λ)^{k−1} e^{-(x/λ)^k} for x ≥ 0. Here:
- k > 0 is the shape parameter.
- λ > 0 is the scale parameter.
3. What is the cumulative distribution function (CDF) of the Weibull distribution?
The Weibull CDF is F(x) = 1 − e^{-(x/λ)^k} for x ≥ 0. This function gives the probability that a random variable X is less than or equal to x:
- It represents the probability of failure before time x.
- It is widely used in survival analysis and reliability engineering.
4. What do the shape and scale parameters mean in the Weibull distribution?
The shape parameter (k) determines the failure rate pattern, while the scale parameter (λ) controls the spread of the distribution. Specifically:
- If k < 1, the failure rate decreases over time.
- If k = 1, the failure rate is constant (exponential case).
- If k > 1, the failure rate increases over time.
- Larger λ values stretch the distribution to the right.
5. How do you calculate the mean of a Weibull distribution?
The mean of a Weibull distribution is μ = λ Γ(1 + 1/k), where Γ is the Gamma function. To compute it:
- Find 1 + 1/k.
- Evaluate the Gamma function Γ at that value.
- Multiply the result by λ.
6. What is the variance of the Weibull distribution?
The variance of a Weibull distribution is Var(X) = λ²[Γ(1 + 2/k) − (Γ(1 + 1/k))²]. It depends on both the shape parameter k and scale parameter λ. This formula is commonly used in reliability and statistical modeling to measure spread in lifetime data.
7. How is the Weibull distribution related to the exponential distribution?
The Weibull distribution reduces to the exponential distribution when the shape parameter k = 1. Substituting k = 1 into the PDF gives:
- f(x) = (1/λ)e^{-x/λ}
8. How do you solve a Weibull probability example?
To find a probability, substitute values into the Weibull CDF formula F(x) = 1 − e^{-(x/λ)^k}. Example: Let k = 2, λ = 5, and x = 3.
- Compute (3/5)² = 0.36.
- Then F(3) = 1 − e^{-0.36}.
- Since e^{-0.36} ≈ 0.6977,
- F(3) ≈ 1 − 0.6977 = 0.3023.
9. What are the main applications of the Weibull distribution?
The Weibull distribution is mainly used to model lifetime and reliability data. Common applications include:
- Reliability engineering (machine failure times)
- Survival analysis in medical statistics
- Wind speed modeling
- Material strength testing
10. What is the hazard function of the Weibull distribution?
The hazard function of the Weibull distribution is h(x) = (k/λ)(x/λ)^{k−1}. It represents the instantaneous failure rate at time x. Depending on k:
- k < 1 → decreasing hazard rate
- k = 1 → constant hazard rate
- k > 1 → increasing hazard rate
