A stick of length 1 is broken at a point chosen uniformly along its length. One piece is used as the length of a rectangle, and the other is used as the width. Find the mean area of a rectangle formed in this way.
What will be an ideal response?
Let U be the length of the part of the stick used as the length of the rectangle. Then the width is 1 ? U, so the
area is U (1 ? U).
Since U ~ U (0, 1), the probability density function of U is f (u) = 1 for 0 < u < 1 and f (u) = 0 otherwise.
The mean area is therefore 
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