handout_week6_b - over the region R z F Z z = Z Z R z f XY...

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Stochastic Signals and Systems Multiple Random Variables Virginia Tech Fall 2008 One Function of Two RVs Given two random variables X and Y and a function g ( x , y ) , we form a new random variable Z as Z = g ( X , Y ) Given the joint pdf f XY ( x , y ) , how does one obtain f Z ( z ) , the pdf of Z ?
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One Function of Two RVs Three methods: 1 By determining the probability F Z ( z ) = P [ Z z ] = P [ g ( X , Y ) z ] . 2 By fixing one of the two random variables, say Y , at a particular one of its possible values y , and then using f Z ( z ) = Z -∞ f Z ( z | y ) f Y ( y ) dy . 3 “Auxiliary” variable. No general statement can be made as to how to decide in a given problem which method is the simplest. In attacking a problem of this kind, it is wise first to sketch contours in the XY -plane along which the function g ( X , Y ) is constant. One Function of Two RVs: Method 1 Calculate the cumulative distribution function of Z = g ( X , Y ) by determining the probability F Z ( z ) = P [ Z z ] = P [ g ( X , Y ) z ] . The event g ( X , Y ) z is represented by a certain region in the XY -plane, which is bounded by the curve g ( x , y ) = z . Call that region R ( z ) . The probability is found by integrating the joint pdf f XY ( x , y
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Unformatted text preview: ) over the region R ( z ) , F Z ( z ) = Z Z R ( z ) f XY ( x , y ) dxdy . One Function of Two RVs: Method 1 Let Z = X + Y . Determine the cdf and pdf f Z ( z ) . (Cont.): The pdf of f Z ( z ) of Z = g ( X , Y ) is obtained by differentiating this cdf with respect to z . Note: Leibnitz rule: d dz Z b ( z ) a ( z ) g ( x , z ) dx = db ( z ) dz g ( b ( z ) , z )-da ( z ) dz g ( a ( z ) , z )+ Z b ( z ) a ( z ) ∂ ∂ z g ( x , z ) dx One Function of Two RVs: Method 1 The random variables X and Y have joint density f X , Y ( x , y ) = ± e-y ≤ x ≤ y < ∞ otherwise Find the probability density of the sum Z = X + Y . One Function of Two RVs: Method 1 Let Z = max ( X , Y ) . Determine the cdf and pdf f Z ( z ) . One Function of Two RVs: Method 1 The random variables X and Y have the joint density f X , Y ( x , y ) = ± x / y ≤ x ≤ y ≤ 2 otherwise Find the density of the product Z = XY ....
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This note was uploaded on 05/05/2010 for the course ECECS 5605 taught by Professor Dasilver during the Fall '08 term at Virginia Tech.

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handout_week6_b - over the region R z F Z z = Z Z R z f XY...

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