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Unformatted text preview: roposition 4.7.4 yields
√ fW,Z (α, β ) = β
α+ √α −1 2α 0 if (α, β ) ∈ A
else. Example 4.7.6 Suppose X and Y are independent N (0, σ 2 ) random variables. View X as a
Y
random point in the u − v plane, and let (R, Θ) denote the polar coordinates of that point. Find
the joint pdf of R and Θ.
Solution: Changing from rectangular coordinates to polar coordinates can be viewed as a map1
ping g from the u − v plane to the set [0, ∞) × [0, 2π ) in the r − θ plane, where r = (u2 + v 2 ) 2 and
θ = tan−1 ( x2 ). The inverse of this mapping is given by
x1
u = r cos(θ)
v = r sin(θ). 4.7. JOINT PDFS OF FUNCTIONS OF RANDOM VARIABLES 149 The joint pdf of X and Y is given by
fX,V (u, v ) = fX (u)fY (v ) = v
1 − u2 +2 2
1 − r22
e 2σ =
e 2σ .
2πσ 2
2πσ 2 The Jacobian of the mapping from (u, v ) to (r, θ) is given by J = ∂r
∂u ∂r
∂v ∂θ
∂u ∂θ
∂v = u
r v
r − rv2 u
r2 . and so det(J ) = 1 . (This corresponds to the well known fact from calculus that dudv = rdrd...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.
 Spring '08
 Zahrn
 The Land

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