ECE313.Lecture35

# ECE313.Lecture35 - ECE 313 Probability with Engineering...

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Functions of Many Random Variables I 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 35 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 40 Random point in the plane ( X , Y ) denotes a random point in the plane The random variable Z is a function of X and Y ; say Z = g( X , Y ) Example: Z = X + Y meaning that outcome ϖ ∈ Ω is mapped onto the number Z ( ϖ ) = X ( ϖ ) + Y ( ϖ ) by Z Questions: Discrete RV Z : What is the pmf of Z ? Continuous RV Z : What is the pdf of Z ?
ECE 313 - Lecture 35 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 40 X and Y are discrete RVs taking on values u 1 , u 2 , …, u n , … and v 1 , v 2 , …, v m , … respectively Z is a discrete RV taking on values in set {g(u i , v j ), 1 ≤ i ≤ n, 1 ≤ j ≤ m} The m × n values need not all be distinct p Z ( α ) = P{ Z = α } = p X , Y (u i ,v j ) 2200 i 2200 j such that g(u i , v j ) = α Discrete random variable Z = g(X,Y)

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ECE 313 - Lecture 35 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 40 Z has value α at all points on the “curve” g(u,v) = α in the u-v plane Simply add up all the probability masses that happen to lie on the “curve” to get p Z ( α ) = P{ Z = α } = p X , Y (u i ,v j ) 2200 i 2200 j such that g(u i , v j ) = α Repeat for all choices of values of Z Check that you have found a valid pmf Discrete random variable Z = g(X,Y)
ECE 313 - Lecture 35 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 40 p Z ( α ) = P{ Z = α } = p X , Y (u i , α – u i ) i An important special case is when X and Y take on integer values only p Z (n) = P{ Z = n} = p X , Y (i, n – i) i A similar result applies whenever the values of X and Y are equally spaced along the axes Example: Sum: Z = X + Y

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ECE 313 - Lecture 35 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 40 p Z (n) = P{ Z = n} = p X , Y (i, n – i) p X , Y (i, n – i) = p Y | X (n–i | i)•p X (i) = q(n–i)•p X (i) p Z (n) = P{ Z = n} = p X + Y (n) = q(n–i)•p X (i) = discrete convolution of q(•) and p X (•), the unconditional pmf of X Conditioned on X = i, P{ X + Y = n | X = i} = P{ Y = n–i | X = i} The convolution shown obtains P{ Z = n} via the theorem of total probability Sum of integer-valued discrete RVs
ECE 313 - Lecture 35 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 40 p Z (n) = P{ Z = n} = p X , Y (i, n – i) If X and Y are independent

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## This note was uploaded on 09/29/2009 for the course ECE 123 taught by Professor Mr.pil during the Spring '09 term at University of Iowa.

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ECE313.Lecture35 - ECE 313 Probability with Engineering...

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