This is part of a short series on the common life data distributions.

The Weibull distribution is both popular and useful. It has some nice features and flexibility that support its popularity. This short article focuses on 7 formulas of the Weibull Distribution.

If you want to know more about fitting a set of data to a distribution, well that is in another article.

It has the essential formulas that you may find useful when answering specific questions. Knowing a distribution’s set of parameters does provide, along with the right formulas, a quick means to answer a wide range of reliability related questions.

## Parameters

The 2-parameter Weibull distribution has a scale and shape parameter. The 3-parameter Weibull includes a location parameter.

The scale parameter is denoted here as eta (*η*). It is defined as the value at the 63.2th percentile and is units of time (*t*).

The shape parameter is denoted here as beta (*β*). It is also known as the slope which is obvious when viewing a linear CDF plot.

One the nice properties of the Weibull distribution is the value of *β* provides some useful information.

- When
*β*is less than 1 the distribution exhibits a decreasing failure rate over time. - When
*β*is equal to 1 the distribution has a constant failure rate (Weibull reduces to an Exponential distribution with*β*=1. - When
*β*is greater than 1 the distribution exhibits an increasing failure rate over time.

PS: I’m using failure rate and hazard rate interchangeably here.

## Probability Density Function (PDF)

When t ≥ 0 then the probability density function formula is:

$$ \displaystyle\large f(t)=\frac{\beta {{t}^{\beta -1}}}{{{\eta }^{\beta }}}{{e}^{-{{\left( \frac{t}{\eta } \right)}^{\beta }}}}$$

A plot of the PDF provides a histogram-like view of the time-to-failure data.

## Cumulative Density Function (CDF)

F(t) is the cumulative probability of failure from time zero till time t. Very handy when estimating the proportion of units that will fail over a warranty period, for example.

$$ \displaystyle\large F(t)=1-{{e}^{-{{\left( \frac{t}{\eta } \right)}^{\beta }}}}$$

## Reliability Function

R(t) is the chance of survival from from time zero till time t. Instead of looking for the proportion that will fail the reliability function determine the proportion that are expected to survive.

$$ \displaystyle\large R(t)={{e}^{-{{\left( \frac{t}{\eta } \right)}^{\beta }}}}$$

## Conditional Survivor Function

The m(x) function provides a means to estimate the chance of survival for a duration beyond some known time, t, over which the item(s) have already survived. What the probability of surviving time x given the item has already survived over time t?

$$ \displaystyle\large m(x)=R(\left. x \right|t)=\frac{R\left( t+x \right)}{R\left( t \right)}={{e}^{\left( \frac{{{t}^{\beta }}-{{\left( t+x \right)}^{\beta }}}{{{\eta }^{\beta }}} \right)}}$$

## Mean Residual Life

This is the cumulative expected life over time *x* given survival till time *t*.

$$ \displaystyle\large u(x)={{e}^{{{\left( \frac{t}{\eta } \right)}^{\beta }}}}\int_{t}^{\infty }{{{e}^{{{\left( \frac{x}{\eta } \right)}^{\beta }}dx}}}$$

## Hazard Rate

This is the instantaneous probability of failure per unit time.

$$ \displaystyle\large h(t)=\frac{\beta }{\eta }{{\left( \frac{t}{\eta } \right)}^{\beta -1}}$$

## Cumulative Hazard Rate

This is the cumluative failure rate from time zero till time t, or the area under the curve described by the hazard rate, *h(t)*.

$$ \displaystyle\large H(t)={{\left( \frac{t}{\eta } \right)}^{\beta }}$$

Johan Eppinga says

Hi Fred

For the last two formulas, do you mean to say:

h(t) = …

H(t) = ..

as oppose to h(x) and H(x) ?

Defining these with an ‘x’ and having no x on the right-hand-side, implies a constant quantity.

Thanks, – JPE

Fred Schenkelberg says

Good catch Jay, thanks for pointing this out. I’ve updated the formulas. cheers, Fred

Sumit Kumar Saini says

Thank you for the Article. It helped me a lot.

Andrew Ghattas says

Given Probability Density Function f(t) and Conditional Survivor Function R(x|t)=R(t+x)R(t), what is the equation for f(x|t)?

Fred Schenkelberg says

Hi Andrew, it would be the derivative of the reliability function. I’ve not worked out this for a conditional situation, maybe you can for this case. cheers, Fred