Trapezoidal distribution

The trapezoidal distribution is a continuous probability distribution on a compact interval.

definition

Example [a, b] = [0.5], [c, d] = [1.3]

The trapezoidal distribution is defined by the probability density function defined on the interval ${\displaystyle \left[a,b\right]}$

${\displaystyle f(x)={\begin{cases}{\frac {2(x-a)}{(b+d-a-c)(c-a)}},&{\text{wenn }}a\leq x<c\\{\frac {2}{b+d-a-c}},&{\text{wenn }}c\leq x\leq d\\{\frac {2(b-x)}{(b+d-a-c)(b-d)}},&{\text{wenn }}d<x\leq b.\end{cases}}}$

The parameters (minimum value), (maximum value) and the interval (most likely values) determine the shape of the trapezoidal distribution . The graph of the density function looks like a trapezoid and gives this distribution its name. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle [c,d]}$ ${\displaystyle a\leq c\leq d\leq b}$

use

The constant uniform distribution defines a range in which an unknown value can be found with the same probability. For points outside this range, the constant uniform distribution means the often unrealistic assumption that their probability is 0. The trapezoidal distribution compensates for this deficiency in that the probabilities for values ​​outside a range of constant probability do not drop abruptly, but linearly to 0.

In the critical path method , the trapezoidal distribution can be used for modeling processes. The parameters and stand for the optimistic or pessimistic process duration, the probable process duration is assumed in the interval . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle [c,d]\subset [a,b]}$

properties

In the case of a trapezoidal random variable with parameters as stated above, the following formulas apply for the expected value and the variance : ${\displaystyle X}$ ${\displaystyle a,b,c,d}$

${\displaystyle \mathrm {E} (X)={\frac {((d+b)^{2}-db)-((a+c)^{2}-ac)}{3(d+b-a-c)}}}$

${\displaystyle \mathrm {Var} (X)={\frac {(d^{2}+b^{2})(d+b)-(a^{2}+c^{2})(a+c)}{6(d+b-a-c)}}-(\mathrm {E} (X))^{2}}$

For the distribution is skewed to the right , ie . For the example shown in the graphic, [a, b] = [0.5] and [c, d] = [1.3] apply . ${\displaystyle c-a<b-d}$ ${\displaystyle \mathrm {E} (X)>m}$ ${\displaystyle \mathrm {E} (X)=48/21>m=9/4}$

Relationship to other distributions

• For , the trapezoidal distribution changes into the triangular distribution . ${\displaystyle c=d}$
• For and there is a constant uniform distribution . ${\displaystyle a=c}$ ${\displaystyle b=d}$
• If , the trapezoidal distribution is symmetrical with respect to the mean , the mean coinciding with the median . The sum of two independent, continuously uniformly distributed random variables is symmetrically trapezoidal. ${\displaystyle b-d=c-a}$ ${\displaystyle \textstyle {\frac {1}{2}}(a+b)={\frac {1}{2}}(c+d)}$

Individual evidence

1. ^ Samuel Kotz, Johan René van Dorp: Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications. World Scientific 2004, ISBN 978-981-256-115-2
2. ↑ L. Lash, Matthew P. Fox, Aliza K. Fink: Applying Quantitative Bias Analysis to Epidemiologic Data , Springer-Verlag 2010, ISBN 978-1-4419-2774-3 , chap. 8: Probabilistic Bias Analysis - Trapezoidal Distribution (Page 121)
3. ^ W. Küpper, K. Lüder, L. Streitferdt: Netzplantechnik , Physica-Verlag HD (1975), ISBN 3-790-80139-9 , Chapter 3.3.3.1: Probability distributions for process and link durations , page 126
4. ↑ Paul. R. Garvey: Probability Methods for Cost Uncertainty Analysis: A Systems Engineering Perspective , Marcel Dekker Inc (1999), ISBN 0-824-78966-0 , Theorem 4.1