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lecturenotesweek3

# lecturenotesweek3 - ISyE 2027 Probability with Applications...

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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 ) 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

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Definition: Let Ω be a sample space. A discrete random variable is a function X : Ω R that takes on a finite number of values a 1 , a 2 , . . . , a n or an infinite number of values of a 1 , a 2 , . . . . Why does this definition help?
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