We present elementary proofs to some formulas given without proofs by K. A. West and J. McClellan in 1993, B. L. Evans and J. H. McClellan in 1994, and J. Cavicchi in 2002, for the calculation of the convolution integrals and sums of piecewise defined functions. Unlike “divide and conquer”...

The exact distributions of the quotients <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$X/Y$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>X</mi> <mo stretchy="false">/</mo> <mi>Y</mi> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$Y/(X+Y)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>Y</mi> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$X$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>X</mi> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$Y$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>Y</mi> </math> </EquationSource> </InlineEquation> are independent and triangularly distributed random variables are obtained. These quotients are useful especially in operations research and reliability...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>