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

The Pareto distribution is a univariate continuous distribution useful when modeling rare events as the survival function slowly decreases as compared to other life distributions. This short article focuses on 7 formulas of the Pareto Continuous Distribution also known as the Pareto distribution of the first kind (there are three kinds, apparently).

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 location parameter, θ, which is greater than zero also sets the lower limit of time, t. It may be referred to as t-minimum.

The shape parameter, α, also greater than zero is sometimes called the Pareto index.

## Probability Density Function (PDF)

When θ ≤ *t < ∞* then the probability density function formula is:

$$ \displaystyle\large f\left( t \right)=\frac{\alpha {{\theta }^{\alpha }}}{{{t}^{\alpha +1}}}$$

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.

$$ \displaystyle\large F\left( t \right)=1-{{\left( \frac{\theta }{t} \right)}^{\alpha }}$$

## 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.

$$ \displaystyle\large R\left( t \right)={{\left( \frac{\theta }{t} \right)}^{\alpha }}$$

## 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*?

$$ \displaystyle\large m\left( t \right)=\frac{{{\left( t+x \right)}^{\alpha }}}{{{t}^{\alpha }}}$$

## Mean Residual Life

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

$$ \displaystyle\large u\left( t \right)=\frac{\int_{t}^{\infty }{R\left( t \right)dx}}{R\left( t \right)}$$

## Hazard Rate

This is the instantaneous probability of failure per unit time.

$$ \displaystyle\large h\left( t \right)=\frac{\alpha }{t}$$

## Cumulative Hazard Rate

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

$$ \displaystyle\large H\left( t \right)=\alpha \ln \left( \frac{t}{\theta } \right)$$

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