Isye 2027

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Unformatted text preview: F , and the probability of such a set is taken to be 2−n . There are also events that don’t depend on a fixed, finite number of observations. For example, let F be the event that an even number of observations is needed until a one is observed. Show that F is an event and then find its probability. Solution: For k ≥ 1, let Ek be the event that the first one occurs on the k th observation. So Ek = {ω : ω1 = ω2 = · · · = ωk−1 = 0 and ωk = 1}. Then Ek depends on only a finite number of observations, so it is an event, and P {Ek } = 2−k . Observe that F = E2 ∪ E4 ∪ E6 ∪ . . . , so F is an event by Axiom E.3. Also, the events E2 , E4 , . . . are mutually exclusive, so by the full version of Axiom P.3 and (1.4): P (F ) = P (E2 ) + P (E4 ) + · · · = 1 4 1+ 1 4 + 1 4 2 + ··· = 1/4 1 =. 1 − (1/4) 3 Example 1.2.5 (Selection of a point in a square) Take Ω to be the square region in the plane, Ω = {(x, y ) : 0 ≤ x < 1, 0 ≤ y < 1}. It can be shown that there is a probability sp...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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