lecturenotesweek3

lecturenotesweek3 - ISyE 2027 Probability with Applications...

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Unformatted text preview: ISyE 2027 Probability with Applications Polly B. He Fall10, Week 3 Random Variables Reading: Section 4.1-4.2. A motivating example: roll a die twice. Ω = { 1 , 2 , 3 , 4 , 5 , 6 } 2 = { ( ω 1 ,ω 2 ) : ω 1 ∈ { 1 ,..., 6 } ,ω 2 ∈ { 1 ,..., 6 }} Suppose we are interested in the sum of the two outcomes. We can define a function S : Ω → R . S ( ω 1 ,ω 2 ) = ω 1 + ω 2 for ( ω 1 ,ω 2 ) ∈ Ω A tabular view of our sample space: ω 1 ω 2 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 What we want to focus: { S = k } = { ( ω 1 ,ω 2 ) ∈ Ω : S ( ω 1 ,ω 2 ) = k } For example, { S = 4 } = { (1 , 3) , (2 , 2) , (3 , 1) } . You can then find the probabilities of P ( S = k ) for k = 1 ,..., 12 by counting the elements, e.g., P ( S = 2) = P ( { (1 , 1) } ) = 1 36 P ( S = 3) = P ( { (1 , 2) , (2 , 1) } ) = 2 36 ... P ( S = 13) = P ( ∅ ) = 0 1 Definition: Let Ω be a sample space. A discrete random variable is a function...
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This note was uploaded on 02/21/2011 for the course ISYE 2027 taught by Professor Zahrn during the Fall '08 term at Georgia Tech.

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lecturenotesweek3 - ISyE 2027 Probability with Applications...

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