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Unformatted text preview: θ
r
for changing integrals in rectangular coordinates to integrals in polar coordinates.) Therefore,
Proposition 4.7.4 yields that for (r, θ) in the support [0, ∞) × [0, 2π ) of (R, Θ),
fR,Θ (r, θ) = rfX,Y (r cos(θ), r sin(θ)) = r − r22
e 2σ .
2πσ 2 Of course fR,Θ (r, θ) = 0 oﬀ the range of the mapping. The joint density factors into a function of r
and a function of θ, so R and Θ are independent. Moreover, R has the Rayleigh distribution with
parameter σ 2 , and Θ is uniformly distributed on [0, 2π ]. The distribution of R here could also be
found using the result of Example 4.6.3, but the analysis here shows that, in addition, for the case
X and Y are independent and both N (0, σ 2 ), the distance R from the origin is independent of the
angle θ.
The result of this example can be used to generate two independent N (0, σ 2 ) random variables,
X and Y , beginning with two independent random variables, A and B , that are uniform on the
interval [0, 1], as follows. Let R = σ...
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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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