Triangular distribution

In probability theory and statistics, the triangular distribution is a continuous probability distribution with the probability density function defined on the interval [a, b]:

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

where a (location), b (scale) and c (shape) are the triangular distribution parameters.

The cumulative distribution function is:

$F(x)=\left\{\begin{matrix} \frac{(x-a)^2}{(b-a)(c-a)} & \mathrm{for\ } a \le x \le c \\ & \\ 1-\frac{(b-x)^2}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b \end{matrix}\right.$

The expected value and variance of a triangular random variable X are:

$\begin{matrix} E(X) &=& \frac{a+b+c}{3} \\ & & \\ \mathrm{Var}(X) &=& \frac{a^2+b^2+c^2-ab-ac-bc}{18} \end{matrix}$

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

$\left.\begin{matrix}f(x) &=& 2x \\ \\ F(x) &=& x^2 \end{matrix}\right\} \mathrm{for\ } 0 \le x \le 1$

$\begin{matrix} E(X) &=& \frac{2}{3} \\ & & \\ \mathrm{Var}(X) &=& \frac{1}{18} \end{matrix}$

Categories: Probability distributions

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