the JOINT SAMPLE SPACE:

Any experimental outcome can be fully described by ( Y, ), where

    Y = distance from the center of the pencil to the closest line (0 y d / 2), and

    = angle of the needle from the horizontal line (0 ).

Since we know that the pencil was randomly thrown, this means that:
    Y is uniformly distributed:

fY(y) = 2 / d ,   0 y d / 2

and ...

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