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11-02 Cont Random Variables (3)

11-02 Cont Random Variables (3) - BTRY 4080 STSCI 4080 Fall...

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BTRY 4080 / STSCI 4080 Fall 2010 297 Sec. 5.7: Distribution of a Function of a Random Variable General problem: Let X be some random variable with cdf F X ( x ) . How do we determine the probability distribution of the transformed random variable Y = g ( X ) , where g ( · ) is some given function? Section 5.7: X is continuous with pdf f X ( · ) . Two cases: g ( · ) is a monotone function on S x g ( · ) is not monotone on S x .
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BTRY 4080 / STSCI 4080 Fall 2010 298 Approach that works in great generality (but which may not always be easy ): Start with CDF of Y : P ( Y y ) = P ( g ( X ) y ) Determine the “relevant” values of y (i.e., those y for which P ( Y y ) is neither zero nor one); For relevant y : express the event { g ( X ) y } as uniontext i { X A i ( y ) } , where A i ( y ) depends on both y and g ( · ) , A i ( y ) A j ( y ) = for i negationslash = j, and A i ( y ) is an interval for every i . Write P ( Y y ) = i P ( X A i ( y )) The last representation is useful precisely because X , hence F X ( · ) , are given.
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BTRY 4080 / STSCI 4080 Fall 2010 299 Example (5.7.7a): Let X U (0 , 1) . Find cdf of Y = X n , n > 0 . Have F Y ( y ) = 0 for y 0 , F Y ( y ) = 1 for y 1 . For any y (0 , 1) , F Y ( y ) = P ( Y y ) = P ( X n y ) = P ( X y 1 /n ) = F X ( y 1 /n ) . Connection to Slide 298 notation: • { X n y } = { g ( X ) y } for g ( u ) = u n . g ( u ) = u n is monotone increasing for u (0 , 1) . A 1 ( y ) = { X y 1 /n } , A j ( y ) = for j 2 . Since X U (0 , 1) , may conclude: F Y ( y ) = 8 > > < > > : 0 y 0 y 1 /n 0 < y < 1 1 y 1 .
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BTRY 4080 / STSCI 4080 Fall 2010 300 Theorem 7.1: Let X be a continuous random variable having probability
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