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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 R1 where the half-width and half-maximum (HWHM) is h1, and similarly R2 where the HWHM is h2. We assume for convenience, without loss of generality, that h2h1.

We now calculate y := x1 + x2, where x1 is drawn from R1 and x2 is drawn from R2. This situation is shown in figure 1.

Figure 1: Trapezoid as the Convolution of Two Rectangles

Here are some of the key relationships:

R1     R2     Trapezoid  LebesgueRemarks
center:     c1     c2     c1 + c2
HWHM:     h1     h2     h2  Lindependent of h1
HWtop:     h1     h2     h2h1
HWbase:     h1     h2     h2 + h1  L1worst-case deviation
stdev:     σ1     σ2     √(σ12 + σ22)  L2Euclidean norm
stdev:
 h1 √3

 h2 √3

 √(h12 + h22) √3

HWHM/stdev:
 √3

 √3

 √3 √(1 + h12/h22)

height:
 1 h1

 1 h2

 1 h2
normalized to unit area

The HWHM of the trapezoid is independent of the width of R1, so long as R1 is narrower than R2.

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).
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