# Triangular distribution

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Parameters Probability density function Cumulative distribution function ${\displaystyle a:~a\in (-\infty ,\infty )}$${\displaystyle b:~a${\displaystyle c:~a\leq c\leq b\,}$ ${\displaystyle a\leq x\leq b\!}$ ${\displaystyle {\begin{cases}0&{\text{for }}x ${\displaystyle {\begin{cases}0&{\text{for }}x\leq a,\\[2pt]{\frac {(x-a)^{2}}{(b-a)(c-a)}}&{\text{for }}a ${\displaystyle {\frac {a+b+c}{3}}}$ ${\displaystyle {\begin{cases}a+{\sqrt {\frac {(b-a)(c-a)}{2}}}&{\text{for }}c\geq {\frac {a+b}{2}},\\[6pt]b-{\sqrt {\frac {(b-a)(b-c)}{2}}}&{\text{for }}c\leq {\frac {a+b}{2}}.\end{cases}}}$ ${\displaystyle c\,}$ ${\displaystyle {\frac {a^{2}+b^{2}+c^{2}-ab-ac-bc}{18}}}$ ${\displaystyle {\frac {{\sqrt {2}}(a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^{2}\!+\!b^{2}\!+\!c^{2}\!-\!ab\!-\!ac\!-\!bc)^{\frac {3}{2}}}}}$ ${\displaystyle -{\frac {3}{5}}}$ ${\displaystyle {\frac {1}{2}}+\ln \left({\frac {b-a}{2}}\right)}$ ${\displaystyle 2{\frac {(b\!-\!c)e^{at}\!-\!(b\!-\!a)e^{ct}\!+\!(c\!-\!a)e^{bt}}{(b-a)(c-a)(b-c)t^{2}}}}$ ${\displaystyle -2{\frac {(b\!-\!c)e^{iat}\!-\!(b\!-\!a)e^{ict}\!+\!(c\!-\!a)e^{ibt}}{(b-a)(c-a)(b-c)t^{2}}}}$

In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, upper limit b and mode c, where a < b and a  c  b.

Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.

Statistics is a branch of mathematics dealing with data collection, organization, analysis, interpretation and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable X is used to denote the outcome of a coin toss, then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails. Examples of random phenomena can include the results of an experiment or survey.

## Special cases

### Mode at a bound

The distribution simplifies when c = a or c = b. For example, if a = 0, b = 1 and c = 1, then the PDF and CDF become:

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable , or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .

${\displaystyle \left.{\begin{array}{rl}f(x)&=2x\\[8pt]F(x)&=x^{2}\end{array}}\right\}{\text{ for }}0\leq x\leq 1}$
{\displaystyle {\begin{aligned}\operatorname {E} (X)&={\frac {2}{3}}\\[8pt]\operatorname {Var} (X)&={\frac {1}{18}}\end{aligned}}}

#### Distribution of the absolute difference of two standard uniform variables

This distribution for a = 0, b = 1 and c = 0 is the distribution of X = |X1  X2|, where X1, X2 are two independent random variables with standard uniform distribution.

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by the two parameters, a and b, which are its minimum and maximum values. The distribution is often abbreviated U(a,b). It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution's support.

{\displaystyle {\begin{aligned}f(x)&=2-2x{\text{ for }}0\leq x<1\\[6pt]F(x)&=2x-x^{2}{\text{ for }}0\leq x<1\\[6pt]E(X)&={\frac {1}{3}}\\[6pt]\operatorname {Var} (X)&={\frac {1}{18}}\end{aligned}}}

### Symmetric triangular distribution

The symmetric case arises when c = (a + b) / 2.

#### Distribution of the sum of two standard uniform variables

This distribution for a = 0, b = 1 and c = 0.5—the mode (i.e., the peak) is exactly in the middle of the interval—corresponds to the distribution of the mean of two standard uniform variables, i.e., the distribution of X = (X1 + X2) / 2, where X1, X2 are two independent random variables with standard uniform distribution in [0, 1]. [1]

${\displaystyle f(x)={\begin{cases}4x&{\text{for }}0\leq x<{\frac {1}{2}}\\4-4x&{\text{for }}{\frac {1}{2}}\leq x\leq 1\end{cases}}}$
${\displaystyle F(x)={\begin{cases}2x^{2}&{\text{for }}0\leq x<{\frac {1}{2}}\\2x^{2}-(2x-1)^{2}&{\text{for }}{\frac {1}{2}}\leq x\leq 1\end{cases}}}$
{\displaystyle {\begin{aligned}E(X)&={\frac {1}{2}}\\[6pt]\operatorname {Var} (X)&={\frac {1}{24}}\end{aligned}}}

## Generating triangular-distributed random variates

Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate

${\displaystyle X={\begin{cases}a+{\sqrt {U(b-a)(c-a)}}&{\text{ for }}0 [2]

where ${\displaystyle F(c)=(c-a)/(b-a)}$, has a triangular distribution with parameters ${\displaystyle a,b}$ and ${\displaystyle c}$. This can be obtained from the cumulative distribution function.

## Use of the distribution

The triangular distribution is typically used as a subjective description of a population for which there is only limited sample data, and especially in cases where the relationship between variables is known but data is scarce (possibly because of the high cost of collection). It is based on a knowledge of the minimum and maximum and an "inspired guess" [3] as to the modal value. For these reasons, the triangle distribution has been called a "lack of knowledge" distribution.

The triangular distribution is therefore often used in business decision making, particularly in simulations. Generally, when not much is known about the distribution of an outcome (say, only its smallest and largest values), it is possible to use the uniform distribution. But if the most likely outcome is also known, then the outcome can be simulated by a triangular distribution. See for example under corporate finance.

### Project management

The triangular distribution, along with the PERT distribution, is also widely used in project management (as an input into PERT and hence critical path method (CPM)) to model events which take place within an interval defined by a minimum and maximum value.

### Audio dithering

The symmetric triangular distribution is commonly used in audio dithering, where it is called TPDF (triangular probability density function).

• Trapezoidal distribution
• Thomas Simpson
• Three-point estimation
• Five-number summary
• Seven-number summary
• Triangular function
• Central limit theorem — The triangle distribution often occurs as a result of adding two uniform random variables together. In other words, the triangle distribution is often (not always) the result of the very first iteration of the central limit theorem summing process (i.e. ${\textstyle n=2}$). In this sense, the triangle distribution can occasionally occur naturally. If this process of summing together more random variables continues (i.e. ${\textstyle n\geq 3}$), then the distribution will become increasingly bell shaped.
• Irwin–Hall distribution — Using an Irwin–Hall distribution is an easy way to generate a triangle distribution.
• Bates distribution — Similar to the Irwin–Hall distribution, but with the values rescaled back into the 0 to 1 range. Useful for computation of a triangle distribution which can subsequently be rescaled and shifted to create other triangle distributions outside of the 0 to 1 range.

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## References

1. Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications. Samuel Kotz and Johan René van Dorp. https://books.google.de/books?id=JO7ICgAAQBAJ&lpg=PA1&dq=chapter%201%20dig%20out%20suitable%20substitutes%20of%20the%20beta%20distribution%20one%20of%20our%20goals&pg=PA3#v=onepage&q&f=false