Peter and Paula both want to cut out a rectangular piece of paper. Because they

Question

Peter and Paula both want to cut out a rectangular piece of paper. Because
they are both probabilists they determine the exact form of the rectangle by
using realizations of a positive rv, say U, as follows. Peter is lazy and generates
just a single realization of this rv; he then cuts out a square that has length and
width equal to this value. Paula likes diversity and generates two independent
realizations of U. She then cuts out a rectangle with width equal to the first
realization and length equal to the second realization.
a. Will the areas cut out by Peter and Paula differ in expectation?
b. If they do, is Peter’s or Paula’s rectangle expected to be larger?

Explanation / Answer

Peter just generates a single realization of U, and so the area of his squarewill be equal to U2. Thus, on average the area of his square is equal toAPeter = E[U2].Paula generates two independent realizations of U for the width and lengthof her rectangle, let us call these realizations U1 and U2. Accordingly, the areaof her rectangle will be equal to U1 ·U2. On average the area of her rectangleequalsAPaula = E[U1 · U2]= E[U1] · E[U2]= E2[U]because U1 and U2 are independent and have the same distribution as U.Therefore,APeter APaula = E[U2] E2[U] = Var[U] 0that is, APeter APaula. On average, the area of Peter’s square is larger thanthat of Paula’s rectangle, even though all lengths and widths of all rectangles(a square is a rectangle) are generated by realizations of the same generic rv,U.

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