# Isye 2027

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Unformatted text preview: = P (no counts in (0,1]) · P (two counts in (1,2]) · P (no counts in (2,3]) λ2 e−λ λ2 −3λ = e−λ e−λ = e. 2! 2 So, by the law of total probability, P (A) = P (B020 ) + P (B111 ) + P (B202 ) λ2 −3λ λ2 e−λ = e + (λe−λ )3 + e−λ 2 2 λ2 e−λ 2 = λ2 λ4 + λ3 + 2 4 e−3λ . (c) By the deﬁnition of conditional probability, P (B020 |A) = = P (B020 A) P (B020 ) = P (A) P (A) λ2 2 + λ2 2 λ3 + λ4 4 = 2 . 2 + 4λ + λ2 Notice that in part (b) we applied the law of total probability to ﬁnd A, and in part (c) we applied the deﬁnition of the conditional probability P (B020 |A). Together, this amounts to application of Bayes rule for ﬁnding P (B020 |A). 86 3.5.3 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES The gamma distribution Let Tr denote the time of the rth count of a Poisson process. Thus, Tr = U1 + · · · + Ur , where U1 , . . . , Ur are independent, exponentially distributed random variables with parameter λ. This characterization of Tr and the method of Section 4.5.2, showing how to ﬁnd the pdf of the sum of independent continuous-type random vari...
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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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