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 h₂ ≥ h₁.

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:

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).