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

The Uniform distribution is a univariate continuous distribution. This short article focuses on 7 formulas of the Uniform Distribution. A common application is as a non-informative prior. Another application is to model a bounded parameter. The uniform distribution also finds application in random number generation.

When α = β = 1, the uniform distribution is a special case of the Beta distribution. The standard uniform distribution has parameters a = 0 and b = 1 resulting in f(t) = 1 within a and b and zero elsewhere.

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 minimum value, a, is the lower bound, and 0 ≤ a < b. The maximum value, b, is the upper bound, and a < b < ∞.

Probability Density Function (PDF)

When a ≤ *t ≤ b* then the probability density function formula is:

$$ \displaystyle\large f\left( t \right)=\left\{ \begin{array}{l}\frac{1}{b-a}\text{ for a}\le \text{t}\le \text{b}\\0\text{ otherwise}\end{array} \right.$$

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.

$$ \displaystyle\large F\left( t \right)=\left\{ \begin{array}{l}0\text{ for }t<a\\\frac{t-a}{b-a}\text{ for }a\le t\le b\\1\text{ for }t>b\end{array} \right.$$

## 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.

$$ \displaystyle\large R\left( t \right)=\left\{ \begin{array}{l}1\text{ for }t<a\\\frac{b-t}{b-a}\text{ for }a\le t\le b\\0\text{ for }t>b\end{array} \right.$$

## 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*?

For *t < a*:

$$ \displaystyle\large m\left( t \right)=\left\{ \begin{array}{l}1\text{ for }t+x<a\\\frac{b-\left( t+x \right)}{b-a}\text{ for }a\le t+x\le b\\0\text{ for }t>b\end{array} \right.$$

For *a ≤ t ≤ b*:

$$ \displaystyle\large m\left( t \right)=\left\{ \begin{array}{l}1\text{ for }t+x<a\\\frac{b-\left( t+x \right)}{b-a}\text{ for }a\le t+x\le b\\0\text{ for }t+x>b\end{array} \right.$$

For *t > b*:

$$ \displaystyle\large m\left( t \right)=0$$

## Mean Residual Life

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

For *t < a*:

$$ \displaystyle\large u\left( t \right)=\frac{1}{2}\left( a+b \right)-t$$

For *a ≤ t ≤ b*:

$$ \displaystyle\large u\left( t \right)=a-t-\frac{{{\left( a-b \right)}^{2}}}{2\left( t-b \right)}$$

For *t > b*:

$$ \displaystyle\large u\left( t \right)=0$$

## Hazard Rate

This is the instantaneous probability of failure per unit time.

$$ \displaystyle\large h\left( t \right)=\left\{ \begin{array}{l}\frac{1}{b-t}\text{ for }a\le t\le b\\0\text{ otherwise}\end{array} \right.$$

## 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)*.

$$ \displaystyle\large H\left( t \right)=\left\{ \begin{array}{*{35}{l}}0\text{ for }t>a \\-\ln \left( \frac{b-t}{b-a} \right)\text{ for }a\le t\le b \\\infty \text{ for }t>b \\\end{array} \right.$$

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