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# A single dot is placed somewhere in the middle of a very long length of yarn at the textile mill. The yarn is then cut into pieces starting from one...

A single dot is placed somewhere in the middle of a very long length of yarn at the textile mill. The yarn is then cut into pieces starting from one end. The lengths of the different pieces are independent, and the length of each piece is distributed according to the same PDF fX(x). Let R be the length of the piece that includes the dot. Determine the expected value of R in each of the following cases.