ECE313.Lecture37

ECE313.Lecture37 - ECE 313 Probability with Engineering...

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Functions of Many Random Variables III Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications
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ECE 313 - Lecture 37 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 36 W = g( X , Y ) and Z = h( X , Y ) are functions of random variables X and Y We have learned how to compute the pmf or pdf or CDF of W and Z individually What is the joint distribution of W and Z ? The joint CDF F W , Z ( α , β ) cannot be obtained from knowledge of the marginal CDFs F W ( α ) and F Z ( β ) … unless, of course, W and Z are known to be independent random variables Two functions of two RVs
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ECE 313 - Lecture 37 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 36 If X and Y are discrete random variables, then so are W = g( X , Y ) and Z = h( X , Y ) Determine the sets of values taken on by W and Z { α k } = {g(u i , v j ) : 1 ≤ i ≤ m, 1 ≤ j ≤ n} { β l } = {h(u i , v j ) : 1 ≤ i ≤ m, 1 ≤ j ≤ n} p W , Z ( α k , β l ) = P{ W = α k , Z = β l } = p X , Y (u i ,v j ) 2200 i 2200 j Two functions of two discrete RVs
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ECE 313 - Lecture 37 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 36 Example M: W = XY, Z = X/Y If W = XY and Z = X / Y , then W and Z are either both positive or both negative Probability masses are in the 1st or 3rd quadrant of the plane with axes α and β For any α > 0, β > 0, W = α , Z = β if X = +√( αβ ), Y = +√( α / β ) or if X = –√( ), Y = –√( α / β ) For any α < 0, β < 0, W = α , Z = β if X = +√( ), Y = –√( α / β ) or if X = –√( ), Y = +√( α / β )
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ECE 313 - Lecture 37 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 36 Example M: hyperbolic coordinates p W , Z ( α , β ) = sum of the probability masses (if any) at the intersection of the hyperbola uv = α and the straight line u/v = β No intersection if sgn( α ) ≠ sgn( β ) v u uv = α α , β > 0 uv = α u v uv = α α , β < 0 uv = α u/v = β
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ECE 313 - Lecture 37 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 36 Example M: conclusion W = XY and Z = X / Y are both positive or both negative p W , Z ( α , β ) = P{ X = +√( αβ ), Y = +√( α / β )} + P{ X = –√( ), Y = –√( α / β ) } = p X , Y (+√( ),+√( α / β )) + p X , Y (–√( ),–√( α / β )) if α > 0, β > 0 Similarly, p W , Z ( α , β ) = p X , Y (+√( ),–√( α / β ))
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ECE 313 - Lecture 37 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 36 Example N: min and max of two RV If W = min{ X , Y } and Z = max{ X , Y }, then W Z always Probability masses lie above the line α = β For any β α , W = α , Z = β if X = α , Y = β or if X = β , Y = α For any β < α , W = α , Z = β is impossible p W , Z ( α , β ) = p X , Y ( α , β ) + p X , Y ( β , α ) if β α p W , Z ( α , β ) = 0 if β < α
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ECE 313 - Lecture 37 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 8 of 36 The folding transformation α β Line α = β u v Line u=v
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ECE313.Lecture37 - ECE 313 Probability with Engineering...

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