Triangular
Probability density function
Cumulative distribution function
Parameters	${\displaystyle a:a\in (-\mathrm{\infty },\mathrm{\infty })}$ ${\displaystyle c:a\le c\le b}$
Support	${\displaystyle a\le x\le b}$
PDF
CDF
Mean	${\displaystyle \frac{a+b+c}{3}}$
Median
Mode	${\displaystyle c}$
Variance	${\displaystyle \frac{a^2+b^2+c^2-ab-ac-bc}{18}}$
Skewness	${\displaystyle \frac{\sqrt{2}(a+b-2c)(2a-b-c)(a-2b+c)}{5(a^2+b^2+c^2-ab-ac-bc{)}^{\frac{3}{2}}}}$
Excess kurtosis	${\displaystyle -\frac{3}{5}}$
Entropy	${\displaystyle \frac{1}{2}+\ln \left(\frac{b-a}{2}\right)}$
MGF	${\displaystyle 2\frac{(b-c)e^{at}-(b-a)e^{ct}+(c-a)e^{bt}}{(b-a)(c-a)(b-c)t^2}}$
CF	${\displaystyle -2\frac{(b-c)e^{iat}-(b-a)e^{ict}+(c-a)e^{ibt}}{(b-a)(c-a)(b-c)t^2}}$

Inprobability theory andstatistics, thetriangular distribution is a continuousprobability distribution with lower limita, upper limitb, and modec, wherea < b anda ≤ c ≤ b.

The distribution simplifies whenc = a orc = b. For example, ifa = 0,b = 1 andc = 1, then thePDF andCDF become:

for${\displaystyle 0\le x\le 1}$.

This distribution fora = 0,b = 1 andc = 0 is the distribution ofX = |X₁ − X₂|, whereX₁,X₂ are two independent random variables with standarduniform distribution.

The symmetric case arises whenc = (a +b) / 2.In this case, an alternate form of the distribution function is:

This distribution fora = 0,b = 1 andc = 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, that is, the distribution ofX = (X₁ + X₂) / 2, whereX₁,X₂ are two independent random variables with standarduniform distribution in [0, 1].^[1] It is the case of theBates distribution for two variables.

Given a random variateU drawn from theuniform distribution in the interval (0, 1), then the variate^[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.

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 inbusiness decision making, particularly insimulations. Generally, when not much is known about thedistribution of an outcome (say, only its smallest and largest values), it is possible to use theuniform distribution. But if the most likely outcome is also known, then the outcome can be simulated by a triangular distribution. See for example undercorporate finance.

The triangular distribution, along with thePERT distribution, is also widely used inproject management (as an input intoPERT and hencecritical path method (CPM)) to model events which take place within an interval defined by a minimum and maximum value.

The symmetric triangular distribution is commonly used inaudio 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 first iteration of the central limit theorem summing process (i.e.$n=2$). In this sense, the triangle distribution can occasionally occur naturally. If this process of summing together more random variables continues (i.e.$n\ge 3$), 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.

• Weisstein, Eric W."Triangular Distribution".MathWorld.
• Triangle Distribution, decisionsciences.org
• Triangular Distribution, brighton-webs.co.uk
• Proof for the variance of triangular distribution, math.stackexchange.com

• Continuous distributions

• CS1 maint: archived copy as title
• Articles with short description
• Short description matches Wikidata