I1 i1 i1 for instance the probability that the outcome

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Unformatted text preview: tisfy ￿∞ ￿ ∞ ∞ ￿ ￿ ￿ P [ai , bi ] = P {[ai , bi ]} = ( b i − ai ) . i=1 i=1 i=1 For instance, the probability that the outcome is in the interval [0, 1/4] is 1/4, the probability the outcome is in the interval [1/3, 1/2] is 1/6, and the probability that the outcome is in either the interval [0, 1/4] or [1/3, 1/2] should be 1/4 + 1/6 = 5/12. That is, P {[0, 1/4] ∪ [1/3, 1/2]} = P {[0, 1/4]} + P {[1/3, 1/2]} = 11 5 +=. 46 12 If P is to be the uniform probability on [0, 1], then it should also be unaffected by shifting. In particular, it should only depend on the length of the interval and not the endpoints themselves. For instance, 1 P {[0, 1/4]} = P {[1/6, 5/12]} = P {[3/4, 1]} = , 4 or, more generally, 1 for every 0 < r ≤ 3/4. 4 Note that if 3/4 < r < 1, then [r, 1/4 + r] is no longer a subset of [0, 1]. But if we allow “wrapping around” then [r, 1/4 + r] might become two disjoint intervals, each a subset of [0, 1], having total length 1/4. For instance, if...
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This document was uploaded on 04/07/2014.

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