Fundamentals of Probability and Statistics for Engineers

Determine the jpdf of Y 1 and Y 2 , defined by

and show that they are independent.

5.29 The jpdf of X and Y is given by

Determine the jpdf of R and and their respective marginal pdfs where

R(^2 Y^2 )1/2 is the vector length and tan^1 (Y/X ) is the phase angle. Are

R and independent?

5.30 Show that an alternate formula for Equation (5.67) is

where

is evaluated atx g^1 (y). Similar alternate forms hold for Equations (5.12), (5.24)

and (5.68).

160 Fundamentals of Probability and Statistics for Engineers

Y 1 ˆX 1 ‡X 2 ; Y 2 ˆ

X 1

Y 1

;

fXY…x;y†ˆ

1

2 ^2

exp



…x^2 ‡y^2 †

2 ^2



; …

1;

1†<…x;y†<…1;1†:



ˆ‡ ˆ^



fY…y†ˆfX‰g^1 …y†ŠjJ^0 j^1 ;

J^0 ˆ

qg 1 …x†

qx 1

qg 1 …x†

qx 2



qg 1 …x†

qxn

... ...

qgn…x†

qx 1

qgn…x†

qx 2



qgn…x†

qxn

(^)
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ˆ^
X