Probability Distributions > Trapezoidal Distribution

## What is a Trapezoidal Distribution?

**Trapezoidal distributions**are in the shape of a trapezoid— a quadrilateral with two parallel and two non-parallel sides. They tend to be a good fit for data that shows fairly rapid growth, a leveling out period, and then fairly rapid decay. All three stages are linear functions: The first stage is increasing with a positive slope; The leveling out period is constant and the final stage is decreasing with a negative slope.

Van Dorp & Kotz define the parameters of the distribution as:

- a =
**minimum**value for the random variable, - b = lower
**mode**(where the constant stage starts), - c = upper
**mode**(where the constant stage ends), - d =
**maximum**value for the random variable, - m =
**growth rate**for the period between a and b, - n =
**decay rate**for the period between c and d, - α =
**boundary ratio**, fx(b)/fx(c) which Kotz & Van Dorp define as “the relative likelihood of capabilities at stage [a,b] and the beginning of the decay stage [c, d].”

Note that the mode of this distribution is not unique; it can take on any value between the lower mode c and the upper mode d.

## PDF of the Trapezoidal Distribution

The Probability Density Function (PDF) for the trapezoidal distribution (From Dorp & Kotz, 2003) is:

Where:

- μ = 2/(d + c – b – a)
^{-1} - a ≤ b ≤ c ≤ d

## CDF

The trapezoidal cumulative distribution function is 0 for x < a and 1 for x ≥ d. Otherwise, it is linear between b and c, and quadratic for a → b and c → d:

## Similar Distributions

- The uniform distribution is a special case of the trapezoidal distribution; It does not have a growth or decay stage so a (the minimum) = c (the lower mode) and d (the maximum) = b (the upper mode).
- The triangular distribution is also a special case of the trapezoidal; It is missing the constant stage so b (the lower mode) = c (the upper mode).

**References:**

Kotz, S. & Dorp. R. (2004). Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications. World Scientific.

Van Dorp, J et. al,. (2004). An Elicitation Procedure for the Generalized Trapezoidal Distribution with a Uniform Central Stage. Decision Analysis, 4(3):156–166, September 2007.

Van Dorp, J. & Kotz, S.(2003) Generalized Trapezoidal Distributions. Metrika, Vol. 58, Issue 1, July.