# lec10 - Geometric pmf Review: X =number of repetitions of...

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Geometric pmf Review: X = number of repetitions of the experiment until we have the ﬁrst success in a Bernoulli experiment. 1. p X ( k ) = P ( X = k ) = (1 - p ) k 1 | {z } k 1 failures · p |{z} success! 2. E [ X ] = 1 p , Var [ X ] = 1 p p 2 3. c.d.f. is: F X ( t ) = P ( X t ) = 1 - (1 - p ) t Example: Examine the following programming statement: Repeat S until B Solution: Assume P ( B = true ) = 0 . 1 and let X be the number of times S is executed. Then, X has a geometric distribution, P ( X = k ) = p X ( k ) = 0 . 9 k 1 · 0 . 1 How often is S executed on average? - What is E [ X ] ? 1

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Geometric pmf Example 2 Example 2. Watch the input queue at the alpha farm for a job that times out. The probability that a job times out is 0.05. Let Y be the number of the ﬁrst job to time out, then Y Geo 0 . 05 . What’s then the probability that the third job times out? P ( Y = 3) = 0 . 95 2 0 . 05 = 0 . 045 Y is less than 3? P ( Y < 3) = P ( Y 2) = 1 - 0 . 95 2 = 0 . 0975 the ﬁrst job to time out is between the third and the seventh? P (3 Y 7) = P ( Y 7) - P ( Y 2) = 1 - 0 . 95 7 - (1 - 0 . 95 2 ) = 0 . 204 2
Geometric pmf Example 2 (cont’d) What are the expected value for Y , what is V ar [ Y ] ?

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## This note was uploaded on 02/01/2012 for the course STAT 330B taught by Professor Zhou during the Spring '11 term at Iowa State.

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lec10 - Geometric pmf Review: X =number of repetitions of...

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