The 3 Parameter Triangle Distribution 4 Formulas

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

The Triangle distribution is univariate continuous distribution. This short article focuses on 4 formulas of the triangle distribution.

The distribution becomes a standard triangle distribution when a = 0, b = 1, thus it has a mean at the $- \sqrt{{c}/{2}\;} -$ and the median is at $- 1-\sqrt{{\left( 1-c \right)}/{2}\;}-$. The distribution becomes a symmetrical triangle distribution when $- c={\left( b-a \right)}/{2}\;-$.

The triangle distribution is used to approximate distributions when the actual distribution is unknown and bounded, often useful for Monte Carlo simulations. Other applications include subjective representation when there is evidence of bounds and a mode, or as a substitution to the beta distribution since it is bounded.

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

This article 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 also the lower bound, and -∞ ≤ a < b.

The maximum value, b, is also the upper bound, and a < b < ∞.

The mode value, c, is the top of the triangle, and a ≤ c ≤ b.

Probability Density Function (PDF)

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

$$ \displaystyle\large f\left( t \right)=\left\{ \begin{array}{l}\frac{2\left( t-a \right)}{\left( b-a \right)\left( c-a \right)}\text{   for }a\le t\le c\\\frac{2\left( b-t \right)}{\left( b-a \right)\left( b-c \right)}\text{   for c}\le t\le b\end{array} \right.$$

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.

$$ \displaystyle\large F\left( t \right)=\left\{ \begin{array}{l}\frac{{{\left( t-a \right)}^{2}}}{\left( b-a \right)\left( c-a \right)}\text{        for }a\le t\le c\\1-\frac{{{\left( b-t \right)}^{2}}}{\left( b-a \right)\left( b-c \right)}\text{   for c}\le t\le b\end{array} \right.$$

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.

$$ \displaystyle\large R\left( t \right)=\left\{ \begin{array}{l}1-\frac{{{\left( t-a \right)}^{2}}}{\left( b-a \right)\left( c-a \right)}\text{    for }a\le t\le c\\\frac{{{\left( b-t \right)}^{2}}}{\left( b-a \right)\left( b-c \right)}\text{         for c}\le t\le b\end{array} \right.$$

Hazard Rate

This is the instantaneous probability of failure per unit time.

$$ \displaystyle\large h\left( t \right)=\left\{ \begin{array}{l}\frac{\left( a-t \right)}{ab-cb+ca+{{t}^{2}}-2at}\text{    for }a\le t\le c\\\frac{2}{b-t}\text{                                for c}\le t<b\end{array} \right.$$