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

The Poisson distribution is a discrete distribution. This short article focuses on 4 formulas of the Poisson Distribution. It is also known as the rare event distribution. It has application in a homogeneous Poisson princess and with renewal theory.

Check the following conditions to verify the process follows a Poisson distribution:

- There is a negligible or impossible chance of two simultaneous events.
- The expected value of vents in a region is proportional to the size of the region.
- The events in non-overlapping regions are independent.

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 shape parameter, μ, is the expected number of event per time period. If modeling failure events, then μ = λt and is the average number of failures in time t.

The parameter μ is always greater than zero.

## Probability Density Function (PDF)

When *k ≥ 0* , *k* is an integer, then the probability density function formula is:

$$ \displaystyle\large f\left( k \right)=\frac{{{\mu }^{k}}}{k!}{{e}^{-\mu }}$$

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.

$$ \displaystyle\large F\left( k \right)={{e}^{-\mu }}\sum\limits_{j=0}^{k}{\frac{{{\mu }^{j}}}{j!}}$$

When μ > 10 you may approximate use a normal distribution:

$$ \displaystyle\large F\left( k \right)\cong \Phi \left( \frac{k+0.5-\mu }{\sqrt{\mu }} \right)$$

## 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.

$$ \displaystyle\large R\left( k \right)=1-F\left( k \right)$$

## Hazard Rate

This is the instantaneous probability of failure per unit time.

$$ \displaystyle\large h\left( k \right)={{\left[ 1+\frac{k!}{\mu }\left( {{e}^{\mu }}-1-\sum\limits_{j=1}^{k}{\frac{{{\mu }^{j}}}{j!}} \right) \right]}^{-1}}$$

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