Let X, Y , and Z be independent N(0, 2). Let _, _, and R
Chapter , Problem 54(choose chapter or problem)
Let X, Y, and Z be independent \(N(0,\sigma^2)\). Let \(\Theta, \Phi\), and R be the corresponding random variables that are the spherical coordinates of (X, Y, Z):
\(x=r \sin \phi \cos \theta\)
\(y=r \sin \phi \sin \theta\)
\(z= r \cos \phi\)
\(0 \leq \phi \leq \pi, \quad 0 \leq \theta \leq 2 \pi\)
Find the joint and marginal densities of \(\Theta, \Phi\), and R.(Hint:\(dx ~dy ~dz = r^2 \sin \phi ~dr ~d \theta ~d \phi\).)
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