lec4 (1)

lec4 (1) - 1. The R.V.s Y = distance from the center of the...

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1. The R.V.’s ± Y = distance from the center of the needle to closest of equidistant parallel lines 0< y < d /2 ± Φ = angle of needle wrt horizontal 0< φ < π
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2. Joint Sample Space y φ φ = π 0 d /2
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3. Joint Probability Distribution ± Want f Y, Φ ( y, φ ) ± Think about that tricky phrase, “At random” ± f ( ) = 1/ π 0 < φ < π ± f Y ( y ) = 2/ d 0 < y < d /2 ± Independence implies f Y, ( y, ) = f ( ) f Y ( y )= constant = 2/( π d )
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4. Working in the Joint Sample Space y φ φ = π 0 Want { y - (l /2) sin( φ)} < 0 or. .. d /2
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4. Working in the Joint Sample Space y φ φ = π 0 y < (l /2) sin φ (l /2) sin d /2
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P = d φ 0 π dy (2 /[ d ]) 0 ( l / 2)sin P = ( l /[ d ]) d sin 0 φ=− ( l /[ d ])( 1 1) P = 2 l /[ d ]
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Bertrand's Paradox
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radius = 1 3 3 3 random chord
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3 3 3 random chord Bertrand's Paradox: What is the probability that a random chord on this circle has length greater than ? 3
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lec4 (1) - 1. The R.V.s Y = distance from the center of the...

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