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Encyclopedia > Triangular distribution
Triangular
Probability density function
Plot of the Triangular PMF
Cumulative distribution function
Plot of the Triangular CMF
Parameters a:~ain (-infty,infty)
b:~b>a,
c:~ale cle b,
Support a le x le b !
Probability density function (pdf) left{ begin{matrix} frac{2(x-a)}{(b-a)(c-a)} & mathrm{for } a le x le c & frac{2(b-x)}{(b-a)(b-c)} & mathrm{for } c < x le b end{matrix} right.
Cumulative distribution function (cdf) left{ begin{matrix} frac{(x-a)^2}{(b-a)(c-a)} & mathrm{for } a le x le c & 1-frac{(b-x)^2}{(b-a)(b-c)} & mathrm{for } c < x le b end{matrix} right.
Mean frac{a+b+c}{3}
Median left{ begin{matrix} a+frac{sqrt{(b-a)(c-a)}}{sqrt{2}} & mathrm{for } c!ge!frac{b!-!a}{2} & b-frac{sqrt{(b-a)(b-c)}}{sqrt{2}} & mathrm{for } c!le!frac{b!-!a}{2} end{matrix} right.
Mode c,
Variance frac{a^2+b^2+c^2-ab-ac-bc}{18}
Skewness 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}}
Kurtosis frac{12}{5}
Entropy frac{1}{2}+lnleft(frac{b-a}{2}right)
mgf 2frac{(b!-!c)e^{at}!-!(b!-!a)e^{ct}!+!(c!-!a)e^{bt}} {(b-a)(c-a)(b-c)t^2}
Char. func. -2frac{(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, mode c and upper limit b. Image File history File links Download high resolution version (1300x975, 72 KB) Triangular distribution PMF File links The following pages link to this file: Triangular distribution ... Image File history File links Download high resolution version (1300x975, 80 KB) Triangular distribution CMF File links The following pages link to this file: Triangular distribution ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... Probability theory is the mathematical study of probability. ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...


f(x|a,b,c)=left{ begin{matrix} frac{2(x-a)}{(b-a)(c-a)} & mathrm{for } a le x le c & frac{2(b-x)}{(b-a)(b-c)} & mathrm{for } c < x le b end{matrix} right.

Contents


Special cases

Two points known

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

left.begin{matrix}f(x) &=& 2x F(x) &=& x^2 end{matrix}right} mathrm{for } 0 le x le 1
begin{matrix} E(X) &=& frac{2}{3} & & mathrm{Var}(X) &=& frac{1}{18} end{matrix}

Distribution of two standard uniform variables

This distribution for a=0, b=1 and c=0.5 is distribution of X = frac{X_1+X_2}{2}, where X1,X2 are two random variables with standard uniform distribution. In mathematics, the uniform distributions are simple probability distributions. ...

F(x)=left{begin{matrix} 2x^2 & mathrm{for }0 le x < frac{1}{2} 1-2(1-x)^2 & mathrm{for }frac{1}{2} le x le 1 end{matrix}right.
begin{matrix} E(X) &=& frac{1}{2} mathrm{Var}(X) &=& frac{1}{24} end{matrix}

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" as to the modal value.


Business simulations

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 one also knows the most likely outcome, then the outcome might be simulated best by a Triangular distribution. Decision making is the cognitive process of selecting a course of action from among multiple alternatives. ... Wooden mechanical horse simulator during WWI. A simulation is an imitation of some real thing, state of affairs, or process. ... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In mathematics, the uniform distributions are simple probability distributions. ...


Project management

The Triangular distribution, along with the Beta distribution, is also widely used in project management (see PERT and CPM) to model events which take place within an interval defined by a minimum and maximum value. In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where α and β are parameters that must be greater than zero and B is the beta function. ... Project management is the discipline of defining and achieving targets while optimizing the use of resources (time, money, people, materials, energy, space, etc) over the course of a project (a set of activities of finite duration). ... PERT network chart for a seven-month project with five milestones (10 through 50) and six activities (A through F). ... In project management, a critical path is the sequence of project network terminal elements with the longest overall duration, determining the shortest time to complete the project. ...


External links and references

  • Triangular Distribution, mathworld.wolfram.com
  • Triangle Distribution, decisionsciences.org
  • Triangular Distribution, brighton-webs.co.uk
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  Results from FactBites:
 
Triangular distribution - Wikipedia, the free encyclopedia (350 words)
In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, mode c and upper limit b.
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).
The Triangular distribution, along with the Beta distribution, is also widely used in project management (see PERT and CPM) to model events which take place within an interval defined by a minimum and maximum value.
Probability distribution - Wikipedia, the free encyclopedia (1449 words)
A probability distribution is a special case of the more general notion of a probability measure, which is a function that assigns probabilities satisfying the Kolmogorov axioms to the measurable sets of a measurable space.
The rectangular distribution is a uniform distribution on [-1/2,1/2].
The triangular distribution on [a, b], a special case of which is the distribution of the sum of two uniformly distributed random variables (the convolution of two uniform distributions).
  More results at FactBites »


 

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