Let X and Y be zero-mean independent Gaussian random variables with common variance o². If Z = √X² + Y² and W = arctan(Y/X) where |W| ≤ π/2, what is fzw (z, w)

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Question 2
Let X and Y be zero-mean independent Gaussian random variables with common
variance o². If Z = √X² + Y² and W =
-
arctan (Y/X) where
|W| ≤ π/2, what is fzw (z, w)
fzw (z, w)
fzw (z,
,w) =
=
fzw (z, w)
=
z+w
πολ
=
e
All other answers are wrong
fzw (z, w)
e
-2²/20²
ποτ
W
πολ
−z²+w²/20²,0 < z <∞, \w\<
e
´,0 < z <∞, |w| </7/2
/20²
,0 < z <∞, |w| < 1/1
- 2,2
2/20²
,0<z<∞0,|w| </
ㅠ
Transcribed Image Text:Question 2 Let X and Y be zero-mean independent Gaussian random variables with common variance o². If Z = √X² + Y² and W = - arctan (Y/X) where |W| ≤ π/2, what is fzw (z, w) fzw (z, w) fzw (z, ,w) = = fzw (z, w) = z+w πολ = e All other answers are wrong fzw (z, w) e -2²/20² ποτ W πολ −z²+w²/20²,0 < z <∞, \w\< e ´,0 < z <∞, |w| </7/2 /20² ,0 < z <∞, |w| < 1/1 - 2,2 2/20² ,0<z<∞0,|w| </ ㅠ
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