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Let X and Y be i.i.d. N (0, 1) random variables. Let (R, θ) be the polar coordinates for thepoint (X, Y ). That is,X = R cos(θ) and Y = R sin(θ) with R > 0, 0 ≤ θ < 2π.(a)  Find the joint PDF of R and θ. (Note that the inverse of the variabletransformation g(X, Y ) = (R, θ) has been done for you.)(b)  Find the marginal PDF of R.(c)  Find the marginal PDF of θ.(Technical note: Despite the fact that (X, Y ) = (0, 0) is excluded from this transformation,the change of variables procedure still works as intended, since the probability that (X, Y ) =exactly (0, 0), or any specific point, is zero for any continuous joint distribution.)

Let X and Y be i.i.d. N (0, 1) random variables. Let (R, θ) be the polar coordinates for thepoint (X, Y ). That is,X = R cos(θ) and Y = R sin(θ) with R > 0, 0 ≤ θ < 2π.(a)  Find the joint PDF of R and θ. (Note that the inverse of the variabletransformation g(X, Y ) = (R, θ) has been done for you.)(b)  Find the marginal PDF of R.(c)  Find the marginal PDF of θ.(Technical note: Despite the fact that (X, Y ) = (0, 0) is excluded from this transformation,the change of variables procedure still works as intended, since the probability that (X, Y ) =exactly (0, 0), or any specific point, is zero for any continuous joint distribution.)

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Let X and Y be i.i.d. N (0, 1) random variables. Let (R, θ) be the polar coordinates for the
point (X, Y ). That is,
X = R cos(θ) and Y = R sin(θ) with R > 0, 0 ≤ θ < 2π.
(a)  Find the joint PDF of R and θ. (Note that the inverse of the variable
transformation g(X, Y ) = (R, θ) has been done for you.)
(b)  Find the marginal PDF of R.
(c)  Find the marginal PDF of θ.
(Technical note: Despite the fact that (X, Y ) = (0, 0) is excluded from this transformation,
the change of variables procedure still works as intended, since the probability that (X, Y ) =
exactly (0, 0), or any specific point, is zero for any continuous joint distribution.) 

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