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Unformatted text preview: EECS 501 2D EXAMPLE OF JACOBIAN METHOD Fall 2001 Given: fx,y (X, Y ) = 6e−(3X +2Y ) for X, Y ≥ 0; 0 otherwise (2D exponential).
Goal: Compute fz,w (Z, W ) for transformation z = x + y and w = x/(x + y ).
1. Compute the inverse transformation of the given one:
z = z (x, y ) = x + y
w = w(x, y ) = x/(x + y ) →Inverse x = x(z, w) = zw
y = y (z, w) = z (1 − w) since w = x/(x + y ) = x/z → x = zw and y = z − x = z − zw = z (1 − w).
2. Compute the Jacobian=determinant of the Jacobian matrix J:
J  = det ∂z
∂x
∂w
∂x ∂z
∂y
∂w
∂y  = det 1 y
(x+y )2 1
−x
(x+y )2 =− 1
x+y  = 1
x+y . since x, y ≥ 0 for this particular fx,y (X, Y ).
3. Substitute inverse transformation into fx,y (X, Y )/J (X, Y ):
fz,w (Z, W ) =
= fx,y (X,Y )
J (X,Y ) X =x(Z,W ),Y =y (Z,W ) fx,y (ZW,Z (1−W ))
1/(ZW +Z (1−W )) as deﬁned above = 6Ze−(3ZW +2Z (1−W )) = 6Ze−Z (W +2) for Z ≥ 0 and 0 ≤ W ≤ 1 since x, y ≥ 0 → 0 ≤ w ≤ 1.
4. If desired, compute marginal pdfs for z and/or w:
fz (Z ) = 6Ze−2Z
fw ( W ) = ∞
0 1 −ZW
e
dW
0 = 6(e−2Z − e−3Z ) for Z ≥ 0. 6Ze−Z (W +2) dZ = 6
(W +2)2 for 0 ≤ W ≤ 1. Check: Both marginal pdfs integrate to 1. Compare to exponential fx (X ), fy (Y ).
Given: fx,y (X, Y ) = 9e−(3X +3Y ) for X, Y ≥ 0; 0 otherwise (2D exponential).
Goal: Compute fz,w (Z, W ) for transformation z = x + y and w = x/(x + y ).
Now get fz,w (Z, W ) = 9Ze−3Z for Z ≥ 0 and 0 ≤ W ≤ 1.
This is a 2nd order Erlang or Gamma pdf in z ; a uniform pdf in w.
Note that z and w are now independent random variables, unlike before. ...
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This note was uploaded on 07/22/2011 for the course EECS 370 taught by Professor Lehman during the Spring '10 term at University of Florida.
 Spring '10
 Lehman

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