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:

f_Y(y) = 2 / d ,   0 y d / 2

f() = 1 / ,   0

Assuming independence, the joint distribution is:

f_Y,(y,) = 2 / ( d ),

0 y d / 2, 0