In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the triangular distribution is a continuous probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b. The probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 is given by
$$
f(x|a,b,c)= \begin{cases}
0 & \mathrm{for\ } x < a, \\
\frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x < c, \\[4pt]
\frac{2}{b-a} & \mathrm{for\ } x = c, \\[4pt]
\frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\[4pt]
0 & \mathrm{for\ } b < x,
\end{cases}


whose cases avoid division by zero if c = a or c = b.
The distribution simplifies when c = a or c = b.