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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