Data605-Week5-HomeWork5-kamath

Problem set 1

Choose independently two numbers B and C at random from the interval [0, 1] with uniform density. Prove that B and C are proper probability distributions.

Note that the point (B,C) is then chosen at random in the unit square.

Find the probability that

1. B + C < 1/2

   B + C = 1 passes through the points (1/2, 0) and (0, 1/2).

   ==> B + C < 1 is the area of the triangle formed in the unit square and the half plane B + C < 0.5

   ==> So, P(B + C) = (Area of triangle with vertices (1/2, 0), (0, 1/2), (0, 0))

   ==> (1/2) * (1/2)(1/2) = 1/8

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2. BC < 1/2

   ==> P(BC < 1/2)

   ==> ∫(x = 1/2 to 1) dx/(2x) = (1/2) ln 2

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3. |B − C| < 1/2

   ==> P(B >= 1/2 + C, C < 1/2 ) + P(C >= 1/2 + B, B < 1/2 )

   ==> P(B > 1/2 , C < 1/2) + P(C > 1/2, B < 1/2)

   ==> B < 1/2 or B > 1/2 occurs half the time, so the probability is 1/2

   ==> C < 1/2 or C > 1/2 occurs half the time, so the probability is 1/2

   ==> 1/2 + 1/2 = 1/4

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4. max{B,C} < 1/2

   ==> Means P(B < 1/2) and P(C < 1/2)

   ==> B < 1/2 occurs half the time, so the probability is 1/2

   ==> C < 1/2 occurs half the time, so the probability is 1/2

   ==> 1/2 x 1/2 = 1/4

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5. min{B,C} < 1/2

   ==> Means P(B < 1/2) or P(C < 1/2)

   ==> B < 1/2 occurs half the time, so the probability is 1/2

   ==> C < 1/2 occurs half the time, so the probability is 1/2

   ==> 1/2 or 1/2 = 1/2

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