[Maths Class Notes] on Weibull Distribution Pdf for Exam

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:

1. Cumulative distribution function

2. Probability density function

3. Shannon entropy

4. Moments

5. 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)

= 5×0.8326

= 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.