Page 177 - Fundamentals of Probability and Statistics for Engineers

160 Fundamentals of Probability and Statistics for Engineers

Determine the jpdf of Y 1 and Y 2 , defined by
X 1
Y 1 ˆ X 1 ‡ X 2 ; Y 2 ˆ ;
Y 1
and show that they are independent.
5.29 The jpdf of X and Y is given by

2
2
1
x ‡ y †
f x; y†ˆ exp ;
1;
1† < x; y† < 1; 1†:
XY 2 2 2 2
Determine the jpdf of R and and their respective marginal pdfs where
2 1/2

(
R ˆ X 2 ‡ Y ) is the vector length and ˆ tan (Y/X ) is the phase angle. Are
1
R and independent?
5.30 Show that an alternate formula for Equation (5.67) is
0
1

1
f y†ˆ f ‰g y†ŠjJ j ;
X
Y
where
qg 1 x† qg 1 x†

qg 1 x†

qx 1 qx 2 qx n

0
J ˆ . . . .
. .
qg n x† qg n x†

qg n x†

qx 1 qx 2 qx n
1

is evaluated at x ˆ g (y). Similar alternate forms hold for Equations (5.12), (5.24)
and (5.68).

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