Trapezoid as the Convolution of Two Rectangles
John Denker
1 Introduction
This is relevant to calculating uncertainties. If you have two inputs
that exhibit rectangular distributions, the output will exhibit a
trapezoidal distribution.
Suppose we have a rectangular distribution R_{1} where the halfwidth
and halfmaximum (HWHM) is h_{1}, and similarly R_{2} where the HWHM
is h_{2}. We assume for convenience, without loss of generality, that
h_{2} ≥ h_{1}.
We now calculate y := x_{1} + x_{2}, where x_{1} is drawn from R_{1} and
x_{2} is drawn from R_{2}. This situation is shown in figure 1.
Figure 1: Trapezoid as the Convolution of Two
Rectangles
Here are some of the key relationships:
  R_{1}   R_{2}   Trapezoid   Lebesgue  Remarks 
center:   c_{1}   c_{2}   c_{1} + c_{2}  
HWHM:   h_{1}   h_{2}   h_{2}   L^{∞}  independent of h_{1} 
HWtop:   h_{1}   h_{2}   h_{2} − h_{1}  
HWbase:   h_{1}   h_{2}   h_{2} + h_{1}   L^{1}  worstcase deviation 
stdev:   σ_{1}   σ_{2}   √(σ_{1}^{2} + σ_{2}^{2})   L^{2}  Euclidean norm 
stdev:       √(h_{1}^{2} + h_{2}^{2}) 

√3 

 
HWHM/stdev:       √3 

√(1 + h_{1}^{2}/h_{2}^{2}) 

 
height:          normalized to unit area 
The HWHM of the trapezoid is independent of the width of R_{1}, so
long as R_{1} is narrower than R_{2}.
The ratio of HWHM to stdev varies quite a bit:

Ratio = √3 ≈ 1.73 (if the
skinnier rectangle is very skinny, so the trapezoid itself is very
nearly rectangular).
 Ratio = √1.5 ≈ 1.22 (if the two rectangles
have equal width, so the trapezoid is triangular).