Weibull Distribution Calculator

Weibull Distribution Calculator

The Weibull distribution with scale parameter \( \lambda > 0 \) and shape parameter \( k > 0 \) has:

PDF: $$ f(x;\lambda,k)=\frac{k}{\lambda}\Bigl(\frac{x}{\lambda}\Bigr)^{k-1}e^{-(x/\lambda)^k} \quad \text{for } x\ge0 $$

CDF: $$ F(x;\lambda,k)=1-e^{-(x/\lambda)^k} \quad \text{for } x\ge0 $$

Step 1: Enter Parameters

Enter a positive value (e.g., 2)

Enter a positive value (e.g., 1.5)

Enter a non-negative value (e.g., 3)

Weibull Distribution: $$ f(x;\lambda,k)=\frac{k}{\lambda}\Bigl(\frac{x}{\lambda}\Bigr)^{k-1}e^{-(x/\lambda)^k} \quad \text{for } x\ge0 $$

Expected Value: $$ \lambda\,\Gamma\Bigl(1+\frac{1}{k}\Bigr) \quad\text{and}\quad Variance: \lambda^2\Bigl[\Gamma\Bigl(1+\frac{2}{k}\Bigr)-\Bigl(\Gamma\Bigl(1+\frac{1}{k}\Bigr)\Bigr)^2\Bigr] $$

The Weibull distribution is a probability distribution used to model failure times, reliability, and survival. It's used in many fields, including engineering, medicine, finance, and more. 

Applications

  • Engineering: Used to model the reliability and lifetime of products, such as ball bearings, capacitors, and vacuum tubes 
  • Medicine: Used to model survival times after a diagnosis, such as cancer 
  • Finance: Used to model financial data, such as high-frequency trading 
  • Extreme events: Used to predict the occurrence of extreme events, such as floods, earthquakes, and high wind speeds 
  • Manufacturing: Used to model infant mortality, or the higher probability of failure at the start of a product's service life 

How it works

The Weibull distribution is a two-parameter family of curves that can generate a variety of probability curve shapes. The shape and scale parameters can be adjusted to fit observed data and predict future reliability. 

History

The Weibull distribution was introduced in 1951 by Swedish engineer and scientist Waloddi Weibull. 

What does a Weibull analysis tell you?

Weibull Analysis provides a comprehensive understanding of how a product performs over time, allowing for numerous ways of putting that information into action. Weibull plots can be examined to gain insight into failure characteristics.

What is an example of a Weibull distribution in real life?

In probability theory and statistics, the Weibull distribution /ˈwaɪbʊl/ is a continuous probability distribution. It models a broad range of random variables, largely in the nature of a time to failure or time between events. Examples are maximum one-day rainfalls and the time a user spends on a web page.

What are the advantages of Weibull distribution?

The Weibull distribution provides a good fit to the data in most cases. Values for the distribution's parameters allow the analyst to understand the failure characteristics of a part/system, providing the basis for cost-saving decisions throughout design, development and deployment.

What is a 3-parameter Weibull distribution?

The Weibull distribution can take a third parameter. The three-parameter Weibull distribution adds a location parameter that is zero in the two-parameter case. If X has a two-parameter Weibull distribution, then Y = X + c has a three-parameter Weibull distribution with the added location parameter c .

How to calculate a Weibull distribution?

You can calculate the Weibull distribution using the probability density function (pdf), cumulative distribution function (cdf), or other functions. You can also use a Weibull distribution calculator or software. 

Probability density function (pdf) 

  • The pdf for the Weibull distribution is used for x ≥ 0
  • The shape parameter is β > 0, and the scale parameter is α > 0

Cumulative distribution function (cdf) 

  • The cdf for the Weibull distribution is F(x; k; λ) = 1 − e−(x/λ)k for x ≥ 0
  • F(x; k; λ) = 0 for x < 0

Other functions The quantile (inverse cumulative distribution) function, The failure rate (or hazard function), and The mean time between failures (MTBF). 

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