1. Let the joint pdf of X and Y be given by fxy(r, y) = C exp + ry where C > 0 is an unknown constant. Hint: this question encourages a geometric approach. In your integrations, you may wish to carry out a change of coordinates so that the new coordinate axes, while still being perpendicular, align well with the area sketched in part (a) below. (a) Sketch the area in the plane R² where fx.y is positive. (b) Do X, Y jointly follow a normal distribution? Justify your answer. (c) Decide whether g(r, y) = C' exp (-2+ry-) defines a pdf on R2 (Note that g is only one factor of fx,y, not the whole of fx,y). If yes, compute C". If no, show that there is no C" > 0 for which it is a pdf. Hint: Consider the change of coordinates w =x + y, z = x - y. (d) Compute the constant C.
1. Let the joint pdf of X and Y be given by fxy(r, y) = C exp + ry where C > 0 is an unknown constant. Hint: this question encourages a geometric approach. In your integrations, you may wish to carry out a change of coordinates so that the new coordinate axes, while still being perpendicular, align well with the area sketched in part (a) below. (a) Sketch the area in the plane R² where fx.y is positive. (b) Do X, Y jointly follow a normal distribution? Justify your answer. (c) Decide whether g(r, y) = C' exp (-2+ry-) defines a pdf on R2 (Note that g is only one factor of fx,y, not the whole of fx,y). If yes, compute C". If no, show that there is no C" > 0 for which it is a pdf. Hint: Consider the change of coordinates w =x + y, z = x - y. (d) Compute the constant C.
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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Transcribed Image Text:1. Let the joint pdf of X and Y be given by
fx.y (r, y) Cexp
1
+ ry
y) 1(0,00) (x + y)
where C > 0 is an unknown constant. Hint: this question encourages a geometric
approach. In your integrations, you may wish to carry out a change of coordinates so
that the new coordinate axes, while still being perpendicular, align well with the area
sketched in part (a) below.
(a) Sketch the area in the plane R? where fx,y is positive.
(b) Do X, Y jointly follow a normal distribution? Justify your answer.
(c) Decide whether g(x, y) = C' exp (-a2 + ry - u) defines a pdf on R2 (Note
that g is only one factor of fxy, not the whole of fx.y). If yes, compute C". If
no, show that there is no C" > 0 for which it is a pdf. Hint: Consider the change
of coordinates w = x +y, z =x - y.
(d) Compute the constant C.
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