A point is picked at random inside the unit diskD = {(x, y) E R² | x² + y² < 1}.Let the random variable R denote the length of the line segment between the origin andthe randomly picked point, and let the random variable e denote the angle betweenthis line segment and the positive horizontal axis (O is measured in radians and so0

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Answered: (b) Compute the marginal distributions…

(b) Compute the marginal distributions of R and O. (c) What is the distribution of O? Are R and O independent random variables? If so, prove it, otherwise, (d) prove they are dependent.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Question

A point is picked at random inside the unit disk
D = {(x, y) E R² | x² + y² < 1}.
Let the random variable R denote the length of the line segment between the origin and
the randomly picked point, and let the random variable e denote the angle between
this line segment and the positive horizontal axis (O is measured in radians and so
0 <O< 2n).
(a) (4 points) Show that the joint distribution of R and O is
if 0 <r < 1, 0 <0 < 2n,
fr,e(r, 0)
otherwise.
Hint: Compute first the CDF FR.e(r, 0). Then, to obtain the PDF, take a derivative
in 0 and followed by a derivative in r:
().
fR,e(r, 0) :
ər (a0FRe(r, 0))
Or \ a0
(b)
Compute the marginal distributions of R and O.
(c)
What is the distribution of O?
(d)
prove they are dependent.
Are R and O independent random variables? If so, prove it, otherwise,

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